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Properties

Label 2-63-1.1-c5-0-9
Degree 22
Conductor 6363
Sign 11
Analytic cond. 10.104110.1041
Root an. cond. 3.178703.17870
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·2-s + 68·4-s + 56·5-s − 49·7-s + 360·8-s + 560·10-s − 232·11-s − 140·13-s − 490·14-s + 1.42e3·16-s + 1.72e3·17-s − 98·19-s + 3.80e3·20-s − 2.32e3·22-s − 1.82e3·23-s + 11·25-s − 1.40e3·26-s − 3.33e3·28-s − 3.41e3·29-s − 7.64e3·31-s + 2.72e3·32-s + 1.72e4·34-s − 2.74e3·35-s − 1.03e4·37-s − 980·38-s + 2.01e4·40-s + 1.79e4·41-s + ⋯
L(s)  = 1  + 1.76·2-s + 17/8·4-s + 1.00·5-s − 0.377·7-s + 1.98·8-s + 1.77·10-s − 0.578·11-s − 0.229·13-s − 0.668·14-s + 1.39·16-s + 1.44·17-s − 0.0622·19-s + 2.12·20-s − 1.02·22-s − 0.718·23-s + 0.00351·25-s − 0.406·26-s − 0.803·28-s − 0.754·29-s − 1.42·31-s + 0.469·32-s + 2.55·34-s − 0.378·35-s − 1.24·37-s − 0.110·38-s + 1.99·40-s + 1.66·41-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: 11
Analytic conductor: 10.104110.1041
Root analytic conductor: 3.178703.17870
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 63, ( :5/2), 1)(2,\ 63,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.9752032934.975203293
L(12)L(\frac12) ≈\approx 4.9752032934.975203293
L(72)L(\frac{7}{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)
bad3 1 1
7 1+p2T 1 + p^{2} T
good2 1−5pT+p5T2 1 - 5 p T + p^{5} T^{2}
5 1−56T+p5T2 1 - 56 T + p^{5} T^{2}
11 1+232T+p5T2 1 + 232 T + p^{5} T^{2}
13 1+140T+p5T2 1 + 140 T + p^{5} T^{2}
17 1−1722T+p5T2 1 - 1722 T + p^{5} T^{2}
19 1+98T+p5T2 1 + 98 T + p^{5} T^{2}
23 1+1824T+p5T2 1 + 1824 T + p^{5} T^{2}
29 1+3418T+p5T2 1 + 3418 T + p^{5} T^{2}
31 1+7644T+p5T2 1 + 7644 T + p^{5} T^{2}
37 1+10398T+p5T2 1 + 10398 T + p^{5} T^{2}
41 1−17962T+p5T2 1 - 17962 T + p^{5} T^{2}
43 1−10880T+p5T2 1 - 10880 T + p^{5} T^{2}
47 1+9324T+p5T2 1 + 9324 T + p^{5} T^{2}
53 1+2262T+p5T2 1 + 2262 T + p^{5} T^{2}
59 1−2730T+p5T2 1 - 2730 T + p^{5} T^{2}
61 1−25648T+p5T2 1 - 25648 T + p^{5} T^{2}
67 1+48404T+p5T2 1 + 48404 T + p^{5} T^{2}
71 1−58560T+p5T2 1 - 58560 T + p^{5} T^{2}
73 1−68082T+p5T2 1 - 68082 T + p^{5} T^{2}
79 1−31784T+p5T2 1 - 31784 T + p^{5} T^{2}
83 1−20538T+p5T2 1 - 20538 T + p^{5} T^{2}
89 1−50582T+p5T2 1 - 50582 T + p^{5} T^{2}
97 1+58506T+p5T2 1 + 58506 T + p^{5} 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

−13.94913286263688147168190464311, −12.96812867206438203949581105958, −12.25188367449539334752303065913, −10.83115965099407741695904204146, −9.639434028993440379376197880531, −7.46641612547275078022193965130, −6.01506616882935937543720922543, −5.29641825413961227450840317371, −3.59013564119637838204717865625, −2.14810415613733623647912509178, 2.14810415613733623647912509178, 3.59013564119637838204717865625, 5.29641825413961227450840317371, 6.01506616882935937543720922543, 7.46641612547275078022193965130, 9.639434028993440379376197880531, 10.83115965099407741695904204146, 12.25188367449539334752303065913, 12.96812867206438203949581105958, 13.94913286263688147168190464311

Graph of the ZZ-function along the critical line