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Properties

Label 12-217e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.044\times 10^{14}$
Sign $1$
Analytic cond. $1.61324\times 10^{-6}$
Root an. cond. $0.329085$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·8-s − 3·9-s − 3·31-s + 3·47-s + 64-s + 3·67-s + 6·72-s + 3·81-s − 3·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·8-s − 3·9-s − 3·31-s + 3·47-s + 64-s + 3·67-s + 6·72-s + 3·81-s − 3·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 31^{6}\)
Sign: $1$
Analytic conductor: \(1.61324\times 10^{-6}\)
Root analytic conductor: \(0.329085\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 31^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09544701282\)
\(L(\frac12)\) \(\approx\) \(0.09544701282\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T^{3} + T^{6} \)
31 \( ( 1 + T + T^{2} )^{3} \)
good2 \( ( 1 + T^{3} + T^{6} )^{2} \)
3 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
5 \( ( 1 + T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T )^{6}( 1 + T )^{6} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
29 \( ( 1 - T )^{6}( 1 + T )^{6} \)
37 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 - T )^{6}( 1 + T )^{6} \)
47 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
53 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
59 \( ( 1 + T^{3} + T^{6} )^{2} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
71 \( ( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
97 \( ( 1 + T^{3} + T^{6} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.06862473446653214153876692839, −6.92004272874006162231180283827, −6.59588867807081214190953247575, −6.46207567997859657789204134876, −6.33320709040343241596457717550, −5.97132307303670086286903608947, −5.82377155207640946766007949885, −5.64654411236283156912412333174, −5.56063569752041873519721147380, −5.47070304278974397902449302267, −5.39971916442881842288417203899, −5.03792675472014003815540025225, −4.76108896630932788461079472667, −4.34909975398010827990160108437, −4.16190392054386340942990041896, −3.98807132369697852645364498630, −3.59751790076284997243670166092, −3.38458605345674831525727285309, −3.21121395812114001929148584739, −3.13446853558491988849746155893, −2.74140242170099618035128208946, −2.39459403238857741081887554421, −2.20142081861122994659996864210, −2.16295260000726395999201023820, −1.21615787733794988885810045727, 1.21615787733794988885810045727, 2.16295260000726395999201023820, 2.20142081861122994659996864210, 2.39459403238857741081887554421, 2.74140242170099618035128208946, 3.13446853558491988849746155893, 3.21121395812114001929148584739, 3.38458605345674831525727285309, 3.59751790076284997243670166092, 3.98807132369697852645364498630, 4.16190392054386340942990041896, 4.34909975398010827990160108437, 4.76108896630932788461079472667, 5.03792675472014003815540025225, 5.39971916442881842288417203899, 5.47070304278974397902449302267, 5.56063569752041873519721147380, 5.64654411236283156912412333174, 5.82377155207640946766007949885, 5.97132307303670086286903608947, 6.33320709040343241596457717550, 6.46207567997859657789204134876, 6.59588867807081214190953247575, 6.92004272874006162231180283827, 7.06862473446653214153876692839

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.