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Properties

Label 2-588-1.1-c7-0-38
Degree 22
Conductor 588588
Sign −1-1
Analytic cond. 183.682183.682
Root an. cond. 13.552913.5529
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s − 100·5-s + 729·9-s + 2.77e3·11-s + 3.29e3·13-s − 2.70e3·15-s − 5.90e3·17-s − 6.64e3·19-s + 1.98e3·23-s − 6.81e4·25-s + 1.96e4·27-s − 2.08e5·29-s + 1.17e5·31-s + 7.48e4·33-s − 3.35e5·37-s + 8.89e4·39-s + 2.65e5·41-s − 9.32e4·43-s − 7.29e4·45-s + 6.57e5·47-s − 1.59e5·51-s − 6.08e5·53-s − 2.77e5·55-s − 1.79e5·57-s + 5.36e5·59-s + 1.79e6·61-s − 3.29e5·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.357·5-s + 1/3·9-s + 0.628·11-s + 0.415·13-s − 0.206·15-s − 0.291·17-s − 0.222·19-s + 0.0339·23-s − 0.871·25-s + 0.192·27-s − 1.58·29-s + 0.710·31-s + 0.362·33-s − 1.08·37-s + 0.240·39-s + 0.601·41-s − 0.178·43-s − 0.119·45-s + 0.923·47-s − 0.168·51-s − 0.561·53-s − 0.224·55-s − 0.128·57-s + 0.339·59-s + 1.01·61-s − 0.148·65-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 183.682183.682
Root analytic conductor: 13.552913.5529
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 588, ( :7/2), −1)(2,\ 588,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
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−p3T 1 - p^{3} T
7 1 1
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

−9.071159579036313765488612344936, −8.307326242178764360213311089958, −7.43797041371714104235115889488, −6.55420609023195689836301393387, −5.48907396589703104925563490100, −4.19082152679254813852217824035, −3.59227678257679080343846517684, −2.34220061013318116921341156866, −1.31369187873096679021517084345, 0, 1.31369187873096679021517084345, 2.34220061013318116921341156866, 3.59227678257679080343846517684, 4.19082152679254813852217824035, 5.48907396589703104925563490100, 6.55420609023195689836301393387, 7.43797041371714104235115889488, 8.307326242178764360213311089958, 9.071159579036313765488612344936

Graph of the ZZ-function along the critical line