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Properties

Label 2-720-1.1-c3-0-26
Degree 22
Conductor 720720
Sign −1-1
Analytic cond. 42.481342.4813
Root an. cond. 6.517776.51777
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 4·7-s − 48·11-s + 2·13-s + 114·17-s − 140·19-s + 72·23-s + 25·25-s − 210·29-s − 272·31-s + 20·35-s − 334·37-s + 198·41-s + 268·43-s + 216·47-s − 327·49-s + 78·53-s − 240·55-s + 240·59-s + 302·61-s + 10·65-s − 596·67-s − 768·71-s − 478·73-s − 192·77-s + 640·79-s − 348·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.215·7-s − 1.31·11-s + 0.0426·13-s + 1.62·17-s − 1.69·19-s + 0.652·23-s + 1/5·25-s − 1.34·29-s − 1.57·31-s + 0.0965·35-s − 1.48·37-s + 0.754·41-s + 0.950·43-s + 0.670·47-s − 0.953·49-s + 0.202·53-s − 0.588·55-s + 0.529·59-s + 0.633·61-s + 0.0190·65-s − 1.08·67-s − 1.28·71-s − 0.766·73-s − 0.284·77-s + 0.911·79-s − 0.460·83-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −1-1
Analytic conductor: 42.481342.4813
Root analytic conductor: 6.517776.51777
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 720, ( :3/2), −1)(2,\ 720,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
5 1−pT 1 - p T
good7 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
11 1+48T+p3T2 1 + 48 T + p^{3} T^{2}
13 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
17 1−114T+p3T2 1 - 114 T + p^{3} T^{2}
19 1+140T+p3T2 1 + 140 T + p^{3} T^{2}
23 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
29 1+210T+p3T2 1 + 210 T + p^{3} T^{2}
31 1+272T+p3T2 1 + 272 T + p^{3} T^{2}
37 1+334T+p3T2 1 + 334 T + p^{3} T^{2}
41 1−198T+p3T2 1 - 198 T + p^{3} T^{2}
43 1−268T+p3T2 1 - 268 T + p^{3} T^{2}
47 1−216T+p3T2 1 - 216 T + p^{3} T^{2}
53 1−78T+p3T2 1 - 78 T + p^{3} T^{2}
59 1−240T+p3T2 1 - 240 T + p^{3} T^{2}
61 1−302T+p3T2 1 - 302 T + p^{3} T^{2}
67 1+596T+p3T2 1 + 596 T + p^{3} T^{2}
71 1+768T+p3T2 1 + 768 T + p^{3} T^{2}
73 1+478T+p3T2 1 + 478 T + p^{3} T^{2}
79 1−640T+p3T2 1 - 640 T + p^{3} T^{2}
83 1+348T+p3T2 1 + 348 T + p^{3} T^{2}
89 1+210T+p3T2 1 + 210 T + p^{3} T^{2}
97 1+1534T+p3T2 1 + 1534 T + p^{3} 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

−9.619491182103650285765017499863, −8.730129701111845440422300778622, −7.82304087992908285675435985494, −7.07599125485653931229090053696, −5.74377227630537620006175915378, −5.29229770631922928062390085809, −3.96918257796561588452124102039, −2.76261014299176375902574960208, −1.64629280193516114942653260711, 0, 1.64629280193516114942653260711, 2.76261014299176375902574960208, 3.96918257796561588452124102039, 5.29229770631922928062390085809, 5.74377227630537620006175915378, 7.07599125485653931229090053696, 7.82304087992908285675435985494, 8.730129701111845440422300778622, 9.619491182103650285765017499863

Graph of the ZZ-function along the critical line