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Properties

Label 2-3131-3131.123-c0-0-2
Degree $2$
Conductor $3131$
Sign $-0.988 + 0.152i$
Analytic cond. $1.56257$
Root an. cond. $1.25002$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.450 + 0.211i)2-s + (−0.479 − 0.579i)4-s + (−0.791 − 1.68i)5-s + (0.340 + 0.362i)7-s + (−0.216 − 0.844i)8-s + (−0.535 − 0.844i)9-s − 0.924i·10-s + (0.0764 + 0.235i)14-s + (−0.0597 + 0.313i)16-s + (−0.0623 − 0.493i)18-s + (−0.273 − 1.43i)19-s + (−0.595 + 1.26i)20-s + (−1.56 + 1.89i)25-s + (0.0469 − 0.371i)28-s + (−0.728 + 0.684i)31-s + (−0.605 + 0.833i)32-s + ⋯
L(s)  = 1  + (0.450 + 0.211i)2-s + (−0.479 − 0.579i)4-s + (−0.791 − 1.68i)5-s + (0.340 + 0.362i)7-s + (−0.216 − 0.844i)8-s + (−0.535 − 0.844i)9-s − 0.924i·10-s + (0.0764 + 0.235i)14-s + (−0.0597 + 0.313i)16-s + (−0.0623 − 0.493i)18-s + (−0.273 − 1.43i)19-s + (−0.595 + 1.26i)20-s + (−1.56 + 1.89i)25-s + (0.0469 − 0.371i)28-s + (−0.728 + 0.684i)31-s + (−0.605 + 0.833i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3131\)    =    \(31 \cdot 101\)
Sign: $-0.988 + 0.152i$
Analytic conductor: \(1.56257\)
Root analytic conductor: \(1.25002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3131} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3131,\ (\ :0),\ -0.988 + 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7665471032\)
\(L(\frac12)\) \(\approx\) \(0.7665471032\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (0.728 - 0.684i)T \)
101 \( 1 + (-0.425 - 0.904i)T \)
good2 \( 1 + (-0.450 - 0.211i)T + (0.637 + 0.770i)T^{2} \)
3 \( 1 + (0.535 + 0.844i)T^{2} \)
5 \( 1 + (0.791 + 1.68i)T + (-0.637 + 0.770i)T^{2} \)
7 \( 1 + (-0.340 - 0.362i)T + (-0.0627 + 0.998i)T^{2} \)
11 \( 1 + (-0.425 + 0.904i)T^{2} \)
13 \( 1 + (-0.0627 - 0.998i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.273 + 1.43i)T + (-0.929 + 0.368i)T^{2} \)
23 \( 1 + (-0.968 - 0.248i)T^{2} \)
29 \( 1 + (0.0627 + 0.998i)T^{2} \)
37 \( 1 + (-0.535 + 0.844i)T^{2} \)
41 \( 1 + (-1.17 - 1.61i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.728 + 0.684i)T^{2} \)
47 \( 1 + (-1.50 - 0.595i)T + (0.728 + 0.684i)T^{2} \)
53 \( 1 + (-0.187 - 0.982i)T^{2} \)
59 \( 1 + (0.946 + 0.180i)T + (0.929 + 0.368i)T^{2} \)
61 \( 1 + (-0.187 + 0.982i)T^{2} \)
67 \( 1 + (0.946 + 1.72i)T + (-0.535 + 0.844i)T^{2} \)
71 \( 1 + (0.328 + 0.180i)T + (0.535 + 0.844i)T^{2} \)
73 \( 1 + (0.968 + 0.248i)T^{2} \)
79 \( 1 + (-0.968 + 0.248i)T^{2} \)
83 \( 1 + (0.968 - 0.248i)T^{2} \)
89 \( 1 + (-0.929 + 0.368i)T^{2} \)
97 \( 1 + (1.03 + 1.24i)T + (-0.187 + 0.982i)T^{2} \)
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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

−8.685335069375610981523247627373, −7.940983496195598743687382420249, −6.92916584372107284782145105176, −5.96107069943622506506370972635, −5.34832372769082244634265876416, −4.61821925548034763884900959904, −4.18043025489922704804372313975, −3.08628277563329032372629232631, −1.43570356312024914988165092831, −0.41901045277649304347289233391, 2.21763594066320985283442638849, 2.87774599286840311292666421111, 3.88863592811570272631684618774, 4.13541696427191987572576023584, 5.42530652321947434258799261823, 6.09921289071078114968956024844, 7.36245149896923440414537071546, 7.56700609799076449736400023637, 8.242027237078813884647579564614, 9.085330172351934959081526763186

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