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Properties

Label 2-3762-209.208-c1-0-63
Degree 22
Conductor 37623762
Sign 0.209+0.977i0.209 + 0.977i
Analytic cond. 30.039730.0397
Root an. cond. 5.480855.48085
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 3.77·5-s + 0.814i·7-s + 8-s − 3.77·10-s + (2.67 − 1.95i)11-s + 2.43·13-s + 0.814i·14-s + 16-s − 3.57i·17-s + (−1.78 + 3.97i)19-s − 3.77·20-s + (2.67 − 1.95i)22-s − 6.28·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.68·5-s + 0.308i·7-s + 0.353·8-s − 1.19·10-s + (0.806 − 0.590i)11-s + 0.676·13-s + 0.217i·14-s + 0.250·16-s − 0.867i·17-s + (−0.408 + 0.912i)19-s − 0.843·20-s + (0.570 − 0.417i)22-s − 1.31·23-s + ⋯

Functional equation

Λ(s)=(3762s/2 ΓC(s) L(s)=((0.209+0.977i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3762s/2 ΓC(s+1/2) L(s)=((0.209+0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37623762    =    2⋅32⋅11⋅192 \cdot 3^{2} \cdot 11 \cdot 19
Sign: 0.209+0.977i0.209 + 0.977i
Analytic conductor: 30.039730.0397
Root analytic conductor: 5.480855.48085
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3762(2089,⋅)\chi_{3762} (2089, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3762, ( :1/2), 0.209+0.977i)(2,\ 3762,\ (\ :1/2),\ 0.209 + 0.977i)

Particular Values

L(1)L(1) ≈\approx 1.6524806631.652480663
L(12)L(\frac12) ≈\approx 1.6524806631.652480663
L(32)L(\frac{3}{2}) 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−T 1 - T
3 1 1
11 1+(−2.67+1.95i)T 1 + (-2.67 + 1.95i)T
19 1+(1.78−3.97i)T 1 + (1.78 - 3.97i)T
good5 1+3.77T+5T2 1 + 3.77T + 5T^{2}
7 1−0.814iT−7T2 1 - 0.814iT - 7T^{2}
13 1−2.43T+13T2 1 - 2.43T + 13T^{2}
17 1+3.57iT−17T2 1 + 3.57iT - 17T^{2}
23 1+6.28T+23T2 1 + 6.28T + 23T^{2}
29 1+6.64T+29T2 1 + 6.64T + 29T^{2}
31 1−2.09iT−31T2 1 - 2.09iT - 31T^{2}
37 1−1.67iT−37T2 1 - 1.67iT - 37T^{2}
41 1−6.59T+41T2 1 - 6.59T + 41T^{2}
43 1+4.68iT−43T2 1 + 4.68iT - 43T^{2}
47 1−4.49T+47T2 1 - 4.49T + 47T^{2}
53 1+11.4iT−53T2 1 + 11.4iT - 53T^{2}
59 1+9.88iT−59T2 1 + 9.88iT - 59T^{2}
61 1+15.1iT−61T2 1 + 15.1iT - 61T^{2}
67 1+2.32iT−67T2 1 + 2.32iT - 67T^{2}
71 1−13.9iT−71T2 1 - 13.9iT - 71T^{2}
73 1+1.29iT−73T2 1 + 1.29iT - 73T^{2}
79 1+5.80T+79T2 1 + 5.80T + 79T^{2}
83 1+7.69iT−83T2 1 + 7.69iT - 83T^{2}
89 1+5.58iT−89T2 1 + 5.58iT - 89T^{2}
97 1+2.59iT−97T2 1 + 2.59iT - 97T^{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

−8.240135120882820376334678708218, −7.61526132723776002076712638393, −6.85633853630423419097400182700, −6.07713809149534451773394822677, −5.32065389640354918578823904383, −4.21837413635892332014295015941, −3.81899006838387817317476979799, −3.19271196860033330199874447371, −1.87112309543056908968465796222, −0.43194598031274989968834818488, 1.09116383056809330029538527979, 2.40453957073597764262169129884, 3.59496129429621326143719504844, 4.17898331541468232412131889811, 4.37117956512073971668868287161, 5.75328254799127259825248396548, 6.40141667502436352285847830487, 7.45389267995886204388835683024, 7.53971705906820343099446389663, 8.590465845592430158701588009578

Graph of the ZZ-function along the critical line