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Properties

Label 2-119-119.118-c0-0-1
Degree $2$
Conductor $119$
Sign $1$
Analytic cond. $0.0593887$
Root an. cond. $0.243698$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 1.61·3-s + 1.61·4-s − 0.618·5-s − 2.61·6-s − 7-s − 8-s + 1.61·9-s + 1.00·10-s + 2.61·12-s + 1.61·14-s − 1.00·15-s − 17-s − 2.61·18-s − 1.00·20-s − 1.61·21-s − 1.61·24-s − 0.618·25-s + 27-s − 1.61·28-s + 1.61·30-s − 0.618·31-s + 32-s + 1.61·34-s + 0.618·35-s + 2.61·36-s + 0.618·40-s + ⋯
L(s)  = 1  − 1.61·2-s + 1.61·3-s + 1.61·4-s − 0.618·5-s − 2.61·6-s − 7-s − 8-s + 1.61·9-s + 1.00·10-s + 2.61·12-s + 1.61·14-s − 1.00·15-s − 17-s − 2.61·18-s − 1.00·20-s − 1.61·21-s − 1.61·24-s − 0.618·25-s + 27-s − 1.61·28-s + 1.61·30-s − 0.618·31-s + 32-s + 1.61·34-s + 0.618·35-s + 2.61·36-s + 0.618·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.0593887\)
Root analytic conductor: \(0.243698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (118, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 119,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4099032861\)
\(L(\frac12)\) \(\approx\) \(0.4099032861\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
17 \( 1 + T \)
good2 \( 1 + 1.61T + T^{2} \)
3 \( 1 - 1.61T + T^{2} \)
5 \( 1 + 0.618T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.61T + T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.61T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 0.618T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.78274557559068410539747469819, −12.81832341078413986102119274275, −11.32903053497152410652726386718, −10.06890896841210859876690174106, −9.307261695204681645728448675397, −8.562806141941940391751284480268, −7.68592225538036914165305710456, −6.75426791784900440825992303891, −3.81385230800766422131085280182, −2.36971271317674985350180148952, 2.36971271317674985350180148952, 3.81385230800766422131085280182, 6.75426791784900440825992303891, 7.68592225538036914165305710456, 8.562806141941940391751284480268, 9.307261695204681645728448675397, 10.06890896841210859876690174106, 11.32903053497152410652726386718, 12.81832341078413986102119274275, 13.78274557559068410539747469819

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