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Properties

Label 2-560-1.1-c5-0-38
Degree 22
Conductor 560560
Sign −1-1
Analytic cond. 89.814989.8149
Root an. cond. 9.477079.47707
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11·3-s − 25·5-s + 49·7-s − 122·9-s − 83·11-s − 83·13-s + 275·15-s − 177·17-s + 2.08e3·19-s − 539·21-s + 3.17e3·23-s + 625·25-s + 4.01e3·27-s − 8.68e3·29-s − 1.63e3·31-s + 913·33-s − 1.22e3·35-s + 4.29e3·37-s + 913·39-s + 2.35e3·41-s − 8.73e3·43-s + 3.05e3·45-s + 3.64e3·47-s + 2.40e3·49-s + 1.94e3·51-s + 3.32e4·53-s + 2.07e3·55-s + ⋯
L(s)  = 1  − 0.705·3-s − 0.447·5-s + 0.377·7-s − 0.502·9-s − 0.206·11-s − 0.136·13-s + 0.315·15-s − 0.148·17-s + 1.32·19-s − 0.266·21-s + 1.24·23-s + 1/5·25-s + 1.05·27-s − 1.91·29-s − 0.305·31-s + 0.145·33-s − 0.169·35-s + 0.516·37-s + 0.0961·39-s + 0.218·41-s − 0.720·43-s + 0.224·45-s + 0.240·47-s + 1/7·49-s + 0.104·51-s + 1.62·53-s + 0.0924·55-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 89.814989.8149
Root analytic conductor: 9.477079.47707
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 560, ( :5/2), −1)(2,\ 560,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
5 1+p2T 1 + p^{2} T
7 1−p2T 1 - p^{2} T
good3 1+11T+p5T2 1 + 11 T + p^{5} T^{2}
11 1+83T+p5T2 1 + 83 T + p^{5} T^{2}
13 1+83T+p5T2 1 + 83 T + p^{5} T^{2}
17 1+177T+p5T2 1 + 177 T + p^{5} T^{2}
19 1−2082T+p5T2 1 - 2082 T + p^{5} T^{2}
23 1−3170T+p5T2 1 - 3170 T + p^{5} T^{2}
29 1+8681T+p5T2 1 + 8681 T + p^{5} T^{2}
31 1+1636T+p5T2 1 + 1636 T + p^{5} T^{2}
37 1−4298T+p5T2 1 - 4298 T + p^{5} T^{2}
41 1−2356T+p5T2 1 - 2356 T + p^{5} T^{2}
43 1+8738T+p5T2 1 + 8738 T + p^{5} T^{2}
47 1−3641T+p5T2 1 - 3641 T + p^{5} T^{2}
53 1−33268T+p5T2 1 - 33268 T + p^{5} T^{2}
59 1−30968T+p5T2 1 - 30968 T + p^{5} T^{2}
61 1−4560T+p5T2 1 - 4560 T + p^{5} T^{2}
67 1+564pT+p5T2 1 + 564 p T + p^{5} T^{2}
71 1−59304T+p5T2 1 - 59304 T + p^{5} T^{2}
73 1+8910T+p5T2 1 + 8910 T + p^{5} T^{2}
79 1+27589T+p5T2 1 + 27589 T + p^{5} T^{2}
83 1+67676T+p5T2 1 + 67676 T + p^{5} T^{2}
89 1−10700T+p5T2 1 - 10700 T + p^{5} T^{2}
97 1−65075T+p5T2 1 - 65075 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

−9.529644084356057352292378550857, −8.663438459030118667228494025299, −7.63990485990199462537927448629, −6.90295383577620248334265382466, −5.60026304184630786692771900917, −5.12226615634501502332815133534, −3.83238861987435720654826558375, −2.67432486975539993859751259551, −1.12775448003691978491480585796, 0, 1.12775448003691978491480585796, 2.67432486975539993859751259551, 3.83238861987435720654826558375, 5.12226615634501502332815133534, 5.60026304184630786692771900917, 6.90295383577620248334265382466, 7.63990485990199462537927448629, 8.663438459030118667228494025299, 9.529644084356057352292378550857

Graph of the ZZ-function along the critical line