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Properties

Label 2-2312-136.59-c0-0-3
Degree 22
Conductor 23122312
Sign 0.880+0.473i0.880 + 0.473i
Analytic cond. 1.153831.15383
Root an. cond. 1.074161.07416
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(2312s/2 ΓC(s) L(s)=((0.880+0.473i) Λ‾(1−s)\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}
Λ(s)=(2312s/2 ΓC(s) L(s)=((0.880+0.473i) Λ‾(1−s)\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: 22
Conductor: 23122312    =    23⋅1722^{3} \cdot 17^{2}
Sign: 0.880+0.473i0.880 + 0.473i
Analytic conductor: 1.153831.15383
Root analytic conductor: 1.074161.07416
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2312(1555,⋅)\chi_{2312} (1555, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2312, ( :0), 0.880+0.473i)(2,\ 2312,\ (\ :0),\ 0.880 + 0.473i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0775389771.077538977
L(12)L(\frac12) ≈\approx 1.0775389771.077538977
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
17 1 1
good3 1+(−0.541+1.30i)T+(−0.707−0.707i)T2 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2}
5 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
7 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
11 1+(−0.541−1.30i)T+(−0.707+0.707i)T2 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2}
13 1+T2 1 + T^{2}
19 1+(−1.41+1.41i)T−iT2 1 + (-1.41 + 1.41i)T - iT^{2}
23 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
29 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
31 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
37 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
41 1+(1.30−0.541i)T+(0.707−0.707i)T2 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+T2 1 + T^{2}
53 1+iT2 1 + iT^{2}
59 1+iT2 1 + iT^{2}
61 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
67 1+T2 1 + T^{2}
71 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
73 1+(1.30+0.541i)T+(0.707+0.707i)T2 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2}
79 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
83 1−iT2 1 - iT^{2}
89 1−T2 1 - T^{2}
97 1+(1.30+0.541i)T+(0.707+0.707i)T2 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line