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Properties

Label 2-252-1.1-c7-0-2
Degree 22
Conductor 252252
Sign 11
Analytic cond. 78.721078.7210
Root an. cond. 8.872488.87248
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 100·5-s − 343·7-s − 2.77e3·11-s − 3.29e3·13-s − 5.90e3·17-s + 6.64e3·19-s − 1.98e3·23-s − 6.81e4·25-s + 2.08e5·29-s − 1.17e5·31-s + 3.43e4·35-s − 3.35e5·37-s + 2.65e5·41-s − 9.32e4·43-s + 6.57e5·47-s + 1.17e5·49-s + 6.08e5·53-s + 2.77e5·55-s + 5.36e5·59-s − 1.79e6·61-s + 3.29e5·65-s + 2.12e6·67-s + 1.19e6·71-s + 1.05e6·73-s + 9.51e5·77-s + 9.98e5·79-s − 3.89e6·83-s + ⋯
L(s)  = 1  − 0.357·5-s − 0.377·7-s − 0.628·11-s − 0.415·13-s − 0.291·17-s + 0.222·19-s − 0.0339·23-s − 0.871·25-s + 1.58·29-s − 0.710·31-s + 0.135·35-s − 1.08·37-s + 0.601·41-s − 0.178·43-s + 0.923·47-s + 1/7·49-s + 0.561·53-s + 0.224·55-s + 0.339·59-s − 1.01·61-s + 0.148·65-s + 0.862·67-s + 0.394·71-s + 0.317·73-s + 0.237·77-s + 0.227·79-s − 0.748·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 11
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 1)(2,\ 252,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 1.3171668561.317166856
L(12)L(\frac12) ≈\approx 1.3171668561.317166856
L(92)L(\frac{9}{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 1
3 1 1
7 1+p3T 1 + p^{3} T
good5 1+4p2T+p7T2 1 + 4 p^{2} T + p^{7} T^{2}
11 1+2774T+p7T2 1 + 2774 T + p^{7} T^{2}
13 1+3294T+p7T2 1 + 3294 T + p^{7} T^{2}
17 1+5900T+p7T2 1 + 5900 T + p^{7} T^{2}
19 1−6644T+p7T2 1 - 6644 T + p^{7} T^{2}
23 1+1982T+p7T2 1 + 1982 T + p^{7} T^{2}
29 1−208106T+p7T2 1 - 208106 T + p^{7} T^{2}
31 1+117792T+p7T2 1 + 117792 T + p^{7} T^{2}
37 1+335686T+p7T2 1 + 335686 T + p^{7} T^{2}
41 1−265488T+p7T2 1 - 265488 T + p^{7} T^{2}
43 1+93292T+p7T2 1 + 93292 T + p^{7} T^{2}
47 1−657516T+p7T2 1 - 657516 T + p^{7} T^{2}
53 1−608718T+p7T2 1 - 608718 T + p^{7} T^{2}
59 1−536120T+p7T2 1 - 536120 T + p^{7} T^{2}
61 1+1797090T+p7T2 1 + 1797090 T + p^{7} T^{2}
67 1−2123176T+p7T2 1 - 2123176 T + p^{7} T^{2}
71 1−1191214T+p7T2 1 - 1191214 T + p^{7} T^{2}
73 1−1056430T+p7T2 1 - 1056430 T + p^{7} T^{2}
79 1−998484T+p7T2 1 - 998484 T + p^{7} T^{2}
83 1+3898004T+p7T2 1 + 3898004 T + p^{7} T^{2}
89 1−4622352T+p7T2 1 - 4622352 T + p^{7} T^{2}
97 1−15287710T+p7T2 1 - 15287710 T + p^{7} 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

−10.72544911227137829316975528226, −9.917145854374944407909832518393, −8.844270400208669519473647219817, −7.82655258821422345976416271711, −6.91545686590752346873852887979, −5.72336211121508169178957754832, −4.59847020299536631694611403790, −3.38301691945411161704719336835, −2.19933198981379827249600908107, −0.56249349651547869118352319677, 0.56249349651547869118352319677, 2.19933198981379827249600908107, 3.38301691945411161704719336835, 4.59847020299536631694611403790, 5.72336211121508169178957754832, 6.91545686590752346873852887979, 7.82655258821422345976416271711, 8.844270400208669519473647219817, 9.917145854374944407909832518393, 10.72544911227137829316975528226

Graph of the ZZ-function along the critical line