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Properties

Label 2-76-19.18-c8-0-1
Degree $2$
Conductor $76$
Sign $0.891 - 0.452i$
Analytic cond. $30.9607$
Root an. cond. $5.56424$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 140. i·3-s − 899.·5-s − 1.72e3·7-s − 1.30e4·9-s + 1.21e4·11-s + 1.55e4i·13-s + 1.25e5i·15-s − 1.45e5·17-s + (1.16e5 − 5.90e4i)19-s + 2.41e5i·21-s + 1.13e5·23-s + 4.18e5·25-s + 9.07e5i·27-s − 1.29e6i·29-s − 5.32e5i·31-s + ⋯
L(s)  = 1  − 1.72i·3-s − 1.43·5-s − 0.717·7-s − 1.98·9-s + 0.829·11-s + 0.543i·13-s + 2.48i·15-s − 1.74·17-s + (0.891 − 0.452i)19-s + 1.24i·21-s + 0.404·23-s + 1.07·25-s + 1.70i·27-s − 1.83i·29-s − 0.576i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.891 - 0.452i$
Analytic conductor: \(30.9607\)
Root analytic conductor: \(5.56424\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :4),\ 0.891 - 0.452i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.389868 + 0.0933414i\)
\(L(\frac12)\) \(\approx\) \(0.389868 + 0.0933414i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-1.16e5 + 5.90e4i)T \)
good3 \( 1 + 140. iT - 6.56e3T^{2} \)
5 \( 1 + 899.T + 3.90e5T^{2} \)
7 \( 1 + 1.72e3T + 5.76e6T^{2} \)
11 \( 1 - 1.21e4T + 2.14e8T^{2} \)
13 \( 1 - 1.55e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.45e5T + 6.97e9T^{2} \)
23 \( 1 - 1.13e5T + 7.83e10T^{2} \)
29 \( 1 + 1.29e6iT - 5.00e11T^{2} \)
31 \( 1 + 5.32e5iT - 8.52e11T^{2} \)
37 \( 1 - 7.97e4iT - 3.51e12T^{2} \)
41 \( 1 - 5.19e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.79e5T + 1.16e13T^{2} \)
47 \( 1 - 2.29e6T + 2.38e13T^{2} \)
53 \( 1 - 2.81e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.69e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.16e7T + 1.91e14T^{2} \)
67 \( 1 - 3.62e7iT - 4.06e14T^{2} \)
71 \( 1 + 2.75e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.71e6T + 8.06e14T^{2} \)
79 \( 1 - 1.73e7iT - 1.51e15T^{2} \)
83 \( 1 - 1.65e7T + 2.25e15T^{2} \)
89 \( 1 + 3.60e6iT - 3.93e15T^{2} \)
97 \( 1 - 5.40e7iT - 7.83e15T^{2} \)
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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

−12.95162559343822290494556711421, −11.64963831109397660724956895529, −11.54453597926444726098945433474, −9.191942477780643385128780140763, −8.051979218615193809522118122944, −7.06424621573849137539204196071, −6.36003552073888989445757762563, −4.14912202421016659329398949048, −2.57869723960626517206353229758, −0.907373421810304308429125255616, 0.16607627143726588730422098666, 3.28555886083838821420216535523, 3.94723583893905260361564910266, 5.11831523311536311606646428812, 6.90488588707035395725524195813, 8.559046584466203187684827638841, 9.358126109867102590175481494658, 10.63773162509062394286547694946, 11.36558213460832215332573119908, 12.51675602427374219914751377299

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