Properties

Label 2-76-76.31-c1-0-3
Degree $2$
Conductor $76$
Sign $0.388 - 0.921i$
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  + (0.835 + 1.14i)2-s + (0.637 + 1.10i)3-s + (−0.603 + 1.90i)4-s + (−1.60 − 2.77i)5-s + (−0.726 + 1.64i)6-s − 1.25i·7-s + (−2.68 + 0.903i)8-s + (0.688 − 1.19i)9-s + (1.82 − 4.14i)10-s + 2.11i·11-s + (−2.48 + 0.548i)12-s + (2.12 + 1.22i)13-s + (1.42 − 1.04i)14-s + (2.04 − 3.53i)15-s + (−3.27 − 2.30i)16-s + (0.765 + 1.32i)17-s + ⋯
L(s)  = 1  + (0.590 + 0.806i)2-s + (0.367 + 0.637i)3-s + (−0.301 + 0.953i)4-s + (−0.717 − 1.24i)5-s + (−0.296 + 0.673i)6-s − 0.472i·7-s + (−0.947 + 0.319i)8-s + (0.229 − 0.397i)9-s + (0.578 − 1.31i)10-s + 0.636i·11-s + (−0.718 + 0.158i)12-s + (0.590 + 0.341i)13-s + (0.381 − 0.279i)14-s + (0.527 − 0.913i)15-s + (−0.817 − 0.575i)16-s + (0.185 + 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.388 - 0.921i$
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: \((2,\ 76,\ (\ :1/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995673 + 0.660570i\)
\(L(\frac12)\) \(\approx\) \(0.995673 + 0.660570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.835 - 1.14i)T \)
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

−14.95703245904219472196616512562, −13.77310033211067831108612016351, −12.62908966856206344545715681851, −11.92539245657084817893211180438, −10.01181326854820807617562941897, −8.720397937825963227004037990437, −7.955650587750011636688199266614, −6.33645921516882084421264142194, −4.49993061453838925603281734883, −3.98613720423025579668921967887, 2.44166774248446138818979681153, 3.78832760409094034986563174189, 5.84660645558549970589018291216, 7.22209497164772111141778225432, 8.571403204877280546696413542908, 10.29355607388337423099509048478, 11.16204669680608795336457096585, 12.08995818589994499565854768091, 13.32614410946287344168838418478, 14.06853311936151899760501817994

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