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Properties

Label 2-76-76.31-c1-0-7
Degree $2$
Conductor $76$
Sign $-0.739 + 0.672i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.570 − 1.29i)2-s + (−0.637 − 1.10i)3-s + (−1.34 + 1.47i)4-s + (−1.60 − 2.77i)5-s + (−1.06 + 1.45i)6-s + 1.25i·7-s + (2.68 + 0.903i)8-s + (0.688 − 1.19i)9-s + (−2.67 + 3.65i)10-s − 2.11i·11-s + (2.48 + 0.548i)12-s + (2.12 + 1.22i)13-s + (1.61 − 0.713i)14-s + (−2.04 + 3.53i)15-s + (−0.359 − 3.98i)16-s + (0.765 + 1.32i)17-s + ⋯
L(s)  = 1  + (−0.403 − 0.915i)2-s + (−0.367 − 0.637i)3-s + (−0.674 + 0.738i)4-s + (−0.717 − 1.24i)5-s + (−0.434 + 0.593i)6-s + 0.472i·7-s + (0.947 + 0.319i)8-s + (0.229 − 0.397i)9-s + (−0.847 + 1.15i)10-s − 0.636i·11-s + (0.718 + 0.158i)12-s + (0.590 + 0.341i)13-s + (0.432 − 0.190i)14-s + (−0.527 + 0.913i)15-s + (−0.0898 − 0.995i)16-s + (0.185 + 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.739 + 0.672i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ -0.739 + 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219630 - 0.568028i\)
\(L(\frac12)\) \(\approx\) \(0.219630 - 0.568028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.570 + 1.29i)T \)
19 \( 1 + (-3.76 + 2.19i)T \)
good3 \( 1 + (0.637 + 1.10i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.60 + 2.77i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.25iT - 7T^{2} \)
11 \( 1 + 2.11iT - 11T^{2} \)
13 \( 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.765 - 1.32i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-7.61 - 4.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.20 + 3.00i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 - 9.97iT - 37T^{2} \)
41 \( 1 + (-1.09 + 0.631i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.04 + 2.91i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.12 + 3.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.18 - 3.57i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.83 - 4.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.80 + 4.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0235 + 0.0408i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.12 - 5.41i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.658 - 1.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.77 - 6.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.84iT - 83T^{2} \)
89 \( 1 + (6.02 + 3.47i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.51 + 4.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.43580994730627728721725549894, −12.82381137144410725252720253891, −11.82865401266075169743663248897, −11.26607158441047197968305318698, −9.373727797071829921253560828468, −8.645630780802033441784910443729, −7.37712462993568658443695382619, −5.35760814191215756760134593172, −3.70681872606563556368126385263, −1.11617401621927805996136514329, 3.86793961139571033658096874209, 5.35920870027863150556404240307, 7.02519332486971638142933579922, 7.64116387022858394721158913585, 9.375563948707857963428711322275, 10.61423470723515621526985400573, 11.02786890518423636830239468118, 12.99657771271881127402266553819, 14.39850890623825995859675369707, 15.01070049284793904857400495220

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