Dirichlet series
| L(s) = 1 | − 4·4-s − 24·5-s − 16·7-s + 16·8-s + 4·9-s + 40·11-s + 8·16-s − 16·17-s − 32·19-s + 96·20-s − 8·23-s + 296·25-s − 16·27-s + 64·28-s + 24·29-s + 32·31-s − 56·32-s + 384·35-s − 16·36-s − 168·37-s − 384·40-s + 96·43-s − 160·44-s − 96·45-s − 80·47-s + 132·49-s + 96·53-s + ⋯ |
| L(s) = 1 | − 4-s − 4.79·5-s − 2.28·7-s + 2·8-s + 4/9·9-s + 3.63·11-s + 1/2·16-s − 0.941·17-s − 1.68·19-s + 24/5·20-s − 0.347·23-s + 11.8·25-s − 0.592·27-s + 16/7·28-s + 0.827·29-s + 1.03·31-s − 7/4·32-s + 10.9·35-s − 4/9·36-s − 4.54·37-s − 9.59·40-s + 2.23·43-s − 3.63·44-s − 2.13·45-s − 1.70·47-s + 2.69·49-s + 1.81·53-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(17^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00211968\) |
| Root analytic conductor: | \(0.680600\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 17^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.1162142978\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1162142978\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( 1 + 16 T + 240 T^{2} - 784 T^{3} - 702 p T^{4} - 784 p^{2} T^{5} + 240 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 2 | \( 1 + p^{2} T^{2} - p^{4} T^{3} + p^{3} T^{4} - 9 p^{3} T^{5} + 37 p^{2} T^{6} - 25 p^{3} T^{7} + 801 T^{8} - 25 p^{5} T^{9} + 37 p^{6} T^{10} - 9 p^{9} T^{11} + p^{11} T^{12} - p^{14} T^{13} + p^{14} T^{14} + p^{16} T^{16} \) |
| 3 | \( 1 - 4 T^{2} + 16 T^{3} + 28 T^{4} - 376 T^{5} + 148 p T^{6} + 8 T^{7} - 6760 T^{8} + 8 p^{2} T^{9} + 148 p^{5} T^{10} - 376 p^{6} T^{11} + 28 p^{8} T^{12} + 16 p^{10} T^{13} - 4 p^{12} T^{14} + p^{16} T^{16} \) | |
| 5 | \( 1 + 24 T + 56 p T^{2} + 2168 T^{3} + 2666 p T^{4} + 15016 p T^{5} + 410024 T^{6} + 2141288 T^{7} + 10795618 T^{8} + 2141288 p^{2} T^{9} + 410024 p^{4} T^{10} + 15016 p^{7} T^{11} + 2666 p^{9} T^{12} + 2168 p^{10} T^{13} + 56 p^{13} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 7 | \( 1 + 16 T + 124 T^{2} + 1296 T^{3} + 10140 T^{4} + 48232 T^{5} + 458172 T^{6} + 3313432 T^{7} + 15111448 T^{8} + 3313432 p^{2} T^{9} + 458172 p^{4} T^{10} + 48232 p^{6} T^{11} + 10140 p^{8} T^{12} + 1296 p^{10} T^{13} + 124 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 - 40 T + 700 T^{2} - 7600 T^{3} + 87412 T^{4} - 1438024 T^{5} + 19499716 T^{6} - 176624752 T^{7} + 1537072952 T^{8} - 176624752 p^{2} T^{9} + 19499716 p^{4} T^{10} - 1438024 p^{6} T^{11} + 87412 p^{8} T^{12} - 7600 p^{10} T^{13} + 700 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 + 784 T^{3} + 9888 T^{4} - 237552 T^{5} + 307328 T^{6} - 20917120 T^{7} + 236770498 T^{8} - 20917120 p^{2} T^{9} + 307328 p^{4} T^{10} - 237552 p^{6} T^{11} + 9888 p^{8} T^{12} + 784 p^{10} T^{13} + p^{16} T^{16} \) | |
| 19 | \( 1 + 32 T + 544 T^{2} + 10608 T^{3} + 192512 T^{4} + 3096432 T^{5} + 96494752 T^{6} + 2593739296 T^{7} + 50479240962 T^{8} + 2593739296 p^{2} T^{9} + 96494752 p^{4} T^{10} + 3096432 p^{6} T^{11} + 192512 p^{8} T^{12} + 10608 p^{10} T^{13} + 544 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 + 8 T + 732 T^{2} + 3584 T^{3} + 513652 T^{4} + 7272312 T^{5} + 445470244 T^{6} + 2689381328 T^{7} + 220561130936 T^{8} + 2689381328 p^{2} T^{9} + 445470244 p^{4} T^{10} + 7272312 p^{6} T^{11} + 513652 p^{8} T^{12} + 3584 p^{10} T^{13} + 732 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 24 T + 684 T^{2} - 83736 T^{3} + 1812294 T^{4} - 39076344 T^{5} + 2881128204 T^{6} - 72099991512 T^{7} + 1161408147714 T^{8} - 72099991512 p^{2} T^{9} + 2881128204 p^{4} T^{10} - 39076344 p^{6} T^{11} + 1812294 p^{8} T^{12} - 83736 p^{10} T^{13} + 684 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 - 32 T + 1988 T^{2} - 84680 T^{3} + 2224116 T^{4} - 107632640 T^{5} + 3131219084 T^{6} - 96339571608 T^{7} + 3769094500984 T^{8} - 96339571608 p^{2} T^{9} + 3131219084 p^{4} T^{10} - 107632640 p^{6} T^{11} + 2224116 p^{8} T^{12} - 84680 p^{10} T^{13} + 1988 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 + 168 T + 13240 T^{2} + 650184 T^{3} + 19962450 T^{4} + 154153272 T^{5} - 24329640440 T^{6} - 1835917350504 T^{7} - 81344425680670 T^{8} - 1835917350504 p^{2} T^{9} - 24329640440 p^{4} T^{10} + 154153272 p^{6} T^{11} + 19962450 p^{8} T^{12} + 650184 p^{10} T^{13} + 13240 p^{12} T^{14} + 168 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 + 5396 T^{2} - 54048 T^{3} + 11503750 T^{4} - 349337248 T^{5} + 11972012020 T^{6} - 1013376724544 T^{7} + 10882038609154 T^{8} - 1013376724544 p^{2} T^{9} + 11972012020 p^{4} T^{10} - 349337248 p^{6} T^{11} + 11503750 p^{8} T^{12} - 54048 p^{10} T^{13} + 5396 p^{12} T^{14} + p^{16} T^{16} \) | |
| 43 | \( 1 - 96 T + 6096 T^{2} - 290832 T^{3} + 11553408 T^{4} - 238904592 T^{5} - 4015115952 T^{6} + 725328019488 T^{7} - 38391385271230 T^{8} + 725328019488 p^{2} T^{9} - 4015115952 p^{4} T^{10} - 238904592 p^{6} T^{11} + 11553408 p^{8} T^{12} - 290832 p^{10} T^{13} + 6096 p^{12} T^{14} - 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 80 T + 3200 T^{2} + 228944 T^{3} + 25698900 T^{4} + 1284226160 T^{5} + 46709290368 T^{6} + 3079489449072 T^{7} + 201735188722534 T^{8} + 3079489449072 p^{2} T^{9} + 46709290368 p^{4} T^{10} + 1284226160 p^{6} T^{11} + 25698900 p^{8} T^{12} + 228944 p^{10} T^{13} + 3200 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 - 96 T + 3844 T^{2} - 250080 T^{3} + 17203208 T^{4} - 878356128 T^{5} + 61303275468 T^{6} - 2343880946208 T^{7} + 44401903321166 T^{8} - 2343880946208 p^{2} T^{9} + 61303275468 p^{4} T^{10} - 878356128 p^{6} T^{11} + 17203208 p^{8} T^{12} - 250080 p^{10} T^{13} + 3844 p^{12} T^{14} - 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 8 T - 1064 T^{2} + 139576 T^{3} - 8160 p T^{4} + 260011896 T^{5} + 38349218008 T^{6} - 28053020936 T^{7} + 74913788078050 T^{8} - 28053020936 p^{2} T^{9} + 38349218008 p^{4} T^{10} + 260011896 p^{6} T^{11} - 8160 p^{9} T^{12} + 139576 p^{10} T^{13} - 1064 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 - 264 T + 33304 T^{2} - 2900248 T^{3} + 200549746 T^{4} - 11914556776 T^{5} + 710898258152 T^{6} - 45325216747576 T^{7} + 2838106764821794 T^{8} - 45325216747576 p^{2} T^{9} + 710898258152 p^{4} T^{10} - 11914556776 p^{6} T^{11} + 200549746 p^{8} T^{12} - 2900248 p^{10} T^{13} + 33304 p^{12} T^{14} - 264 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 30264 T^{2} + 421009100 T^{4} - 3513162425416 T^{6} + 19237575139662822 T^{8} - 3513162425416 p^{4} T^{10} + 421009100 p^{8} T^{12} - 30264 p^{12} T^{14} + p^{16} T^{16} \) | |
| 71 | \( 1 - 32 T - 2196 T^{2} + 376992 T^{3} - 40999012 T^{4} + 1392153480 T^{5} + 50862865468 T^{6} - 10985693543800 T^{7} + 1117214885242200 T^{8} - 10985693543800 p^{2} T^{9} + 50862865468 p^{4} T^{10} + 1392153480 p^{6} T^{11} - 40999012 p^{8} T^{12} + 376992 p^{10} T^{13} - 2196 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 - 24 T + 3500 T^{2} + 527224 T^{3} - 252666 T^{4} - 799621288 T^{5} + 103501902988 T^{6} + 4216386556104 T^{7} - 1450386823086462 T^{8} + 4216386556104 p^{2} T^{9} + 103501902988 p^{4} T^{10} - 799621288 p^{6} T^{11} - 252666 p^{8} T^{12} + 527224 p^{10} T^{13} + 3500 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 + 96 T - 11292 T^{2} - 1308856 T^{3} + 36577844 T^{4} + 2295987344 T^{5} - 542602414484 T^{6} + 12994739703176 T^{7} + 6454474657365240 T^{8} + 12994739703176 p^{2} T^{9} - 542602414484 p^{4} T^{10} + 2295987344 p^{6} T^{11} + 36577844 p^{8} T^{12} - 1308856 p^{10} T^{13} - 11292 p^{12} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 + 88 T + 12840 T^{2} - 772808 T^{3} - 70014880 T^{4} - 15998431560 T^{5} + 276363214376 T^{6} + 33752671318936 T^{7} + 12884842311213474 T^{8} + 33752671318936 p^{2} T^{9} + 276363214376 p^{4} T^{10} - 15998431560 p^{6} T^{11} - 70014880 p^{8} T^{12} - 772808 p^{10} T^{13} + 12840 p^{12} T^{14} + 88 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 288 T + 41472 T^{2} - 4656976 T^{3} + 519733472 T^{4} - 56560463824 T^{5} + 5578739762816 T^{6} - 499331992822432 T^{7} + 43753371662175234 T^{8} - 499331992822432 p^{2} T^{9} + 5578739762816 p^{4} T^{10} - 56560463824 p^{6} T^{11} + 519733472 p^{8} T^{12} - 4656976 p^{10} T^{13} + 41472 p^{12} T^{14} - 288 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 + 344 T + 70152 T^{2} + 11174248 T^{3} + 1407887122 T^{4} + 151612370328 T^{5} + 14250743174520 T^{6} + 12889843831272 p T^{7} + 12417275647714 p^{2} T^{8} + 12889843831272 p^{3} T^{9} + 14250743174520 p^{4} T^{10} + 151612370328 p^{6} T^{11} + 1407887122 p^{8} T^{12} + 11174248 p^{10} T^{13} + 70152 p^{12} T^{14} + 344 p^{14} T^{15} + p^{16} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.507224013604162901297284423054, −9.321395325040976500046796970789, −9.180973547802950940891479458000, −8.740907734124751232301824231597, −8.518022812927411086323442658732, −8.477671329738997284771710937247, −8.471639006345098481783554042851, −8.196592445453276961473738077109, −7.74624421637050607435966712129, −7.58341915264030266880539906037, −7.30185644386452081727377037083, −6.92007369065094415195632223835, −6.81275856473775933151646775250, −6.73993307604006132933322756889, −6.66682316665691600129025257471, −6.44030584999493844216701907112, −5.51390208424260504286822849001, −5.34839051120172782998759335779, −4.66648495643853188972415367283, −4.34969685742836502956987770253, −4.08600961077270380115859267313, −3.99302150832464263260933263353, −3.97512048985636004811259134783, −3.72195384313266909951717340086, −3.19692169658130185197820665222, 3.19692169658130185197820665222, 3.72195384313266909951717340086, 3.97512048985636004811259134783, 3.99302150832464263260933263353, 4.08600961077270380115859267313, 4.34969685742836502956987770253, 4.66648495643853188972415367283, 5.34839051120172782998759335779, 5.51390208424260504286822849001, 6.44030584999493844216701907112, 6.66682316665691600129025257471, 6.73993307604006132933322756889, 6.81275856473775933151646775250, 6.92007369065094415195632223835, 7.30185644386452081727377037083, 7.58341915264030266880539906037, 7.74624421637050607435966712129, 8.196592445453276961473738077109, 8.471639006345098481783554042851, 8.477671329738997284771710937247, 8.518022812927411086323442658732, 8.740907734124751232301824231597, 9.180973547802950940891479458000, 9.321395325040976500046796970789, 9.507224013604162901297284423054
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.