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Properties

Label 2-2312-136.59-c0-0-3
Degree $2$
Conductor $2312$
Sign $0.880 + 0.473i$
Analytic cond. $1.15383$
Root an. cond. $1.07416$
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  + (−0.707 + 0.707i)2-s + (0.541 − 1.30i)3-s − 1.00i·4-s + (0.541 + 1.30i)6-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.541 + 1.30i)11-s + (−1.30 − 0.541i)12-s − 1.00·16-s + 1.00·18-s + (1.41 − 1.41i)19-s + (−1.30 − 0.541i)22-s + (1.30 − 0.541i)24-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + 2·33-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.541 − 1.30i)3-s − 1.00i·4-s + (0.541 + 1.30i)6-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.541 + 1.30i)11-s + (−1.30 − 0.541i)12-s − 1.00·16-s + 1.00·18-s + (1.41 − 1.41i)19-s + (−1.30 − 0.541i)22-s + (1.30 − 0.541i)24-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + 2·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.880 + 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077538977\)
\(L(\frac12)\) \(\approx\) \(1.077538977\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 \)
good3 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
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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

−8.990925396629051595536406652037, −8.258679943539914262932163886708, −7.28094732821319805000857610502, −7.17678630422060205574388188626, −6.49622144435051606131911562509, −5.35185478267536580066357170319, −4.58109093242638849126710593843, −3.00556307929153473232909537629, −1.95312524339242533792463374034, −1.11802801642614217870563074250, 1.25580621749712773381290629238, 2.78036342817622709384871337151, 3.47604866437947015899334751862, 3.99791724622508570879794814397, 5.05622781685907249182612469876, 6.07202079883697532532805514282, 7.21256402714524788870223068893, 8.182784706772219772736733963901, 8.680812225722410760657951309823, 9.287529051946583893873066094909

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