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Properties

Label 2-3131-3131.526-c0-0-1
Degree $2$
Conductor $3131$
Sign $0.991 + 0.128i$
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  + (−0.742 + 0.614i)2-s + (−0.0134 + 0.0703i)4-s + (0.929 − 1.12i)5-s + (0.961 + 0.0604i)7-s + (−0.497 − 0.904i)8-s + (0.425 − 0.904i)9-s + 1.40i·10-s + (−0.751 + 0.545i)14-s + (0.858 + 0.339i)16-s + (0.239 + 0.933i)18-s + (−0.116 + 0.0462i)19-s + (0.0665 + 0.0804i)20-s + (−0.210 − 1.10i)25-s + (−0.0171 + 0.0668i)28-s + (−0.0627 + 0.998i)31-s + (0.136 − 0.0441i)32-s + ⋯
L(s)  = 1  + (−0.742 + 0.614i)2-s + (−0.0134 + 0.0703i)4-s + (0.929 − 1.12i)5-s + (0.961 + 0.0604i)7-s + (−0.497 − 0.904i)8-s + (0.425 − 0.904i)9-s + 1.40i·10-s + (−0.751 + 0.545i)14-s + (0.858 + 0.339i)16-s + (0.239 + 0.933i)18-s + (−0.116 + 0.0462i)19-s + (0.0665 + 0.0804i)20-s + (−0.210 − 1.10i)25-s + (−0.0171 + 0.0668i)28-s + (−0.0627 + 0.998i)31-s + (0.136 − 0.0441i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3131\)    =    \(31 \cdot 101\)
Sign: $0.991 + 0.128i$
Analytic conductor: \(1.56257\)
Root analytic conductor: \(1.25002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3131} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3131,\ (\ :0),\ 0.991 + 0.128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.098511824\)
\(L(\frac12)\) \(\approx\) \(1.098511824\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (0.0627 - 0.998i)T \)
101 \( 1 + (-0.637 + 0.770i)T \)
good2 \( 1 + (0.742 - 0.614i)T + (0.187 - 0.982i)T^{2} \)
3 \( 1 + (-0.425 + 0.904i)T^{2} \)
5 \( 1 + (-0.929 + 1.12i)T + (-0.187 - 0.982i)T^{2} \)
7 \( 1 + (-0.961 - 0.0604i)T + (0.992 + 0.125i)T^{2} \)
11 \( 1 + (-0.637 - 0.770i)T^{2} \)
13 \( 1 + (0.992 - 0.125i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.116 - 0.0462i)T + (0.728 - 0.684i)T^{2} \)
23 \( 1 + (-0.876 - 0.481i)T^{2} \)
29 \( 1 + (-0.992 + 0.125i)T^{2} \)
37 \( 1 + (0.425 + 0.904i)T^{2} \)
41 \( 1 + (0.238 + 0.0774i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.0627 + 0.998i)T^{2} \)
47 \( 1 + (-0.450 - 0.423i)T + (0.0627 + 0.998i)T^{2} \)
53 \( 1 + (-0.929 + 0.368i)T^{2} \)
59 \( 1 + (-0.621 + 1.57i)T + (-0.728 - 0.684i)T^{2} \)
61 \( 1 + (-0.929 - 0.368i)T^{2} \)
67 \( 1 + (-0.621 - 0.394i)T + (0.425 + 0.904i)T^{2} \)
71 \( 1 + (0.996 + 1.57i)T + (-0.425 + 0.904i)T^{2} \)
73 \( 1 + (0.876 + 0.481i)T^{2} \)
79 \( 1 + (-0.876 + 0.481i)T^{2} \)
83 \( 1 + (0.876 - 0.481i)T^{2} \)
89 \( 1 + (0.728 - 0.684i)T^{2} \)
97 \( 1 + (-0.115 + 0.607i)T + (-0.929 - 0.368i)T^{2} \)
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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

−8.723754198908527852445134463391, −8.343843839002533793913504997145, −7.47725251704315350213077233309, −6.68625963421825118547772615073, −5.95549707413338181386251967921, −5.09979417680836927447793246669, −4.36241961184830010262568828757, −3.33582149146906472889635144165, −1.85639703263106781388667530953, −0.971381047154830326558514696756, 1.42334413189425251298201022868, 2.17481754822345500894598954662, 2.76180581689413392513322132317, 4.19109034947748872102187934287, 5.20775020129385576738302356951, 5.77246010163425232556027895657, 6.71334683292921165246968078228, 7.55009760650836306199594897384, 8.221281141553986244465554623528, 9.025127435768348364212700791534

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