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Properties

Label 2-56784-1.1-c1-0-4
Degree 22
Conductor 5678456784
Sign 11
Analytic cond. 453.422453.422
Root an. cond. 21.293721.2937
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 2·11-s + 15-s + 19-s + 21-s − 3·23-s − 4·25-s − 27-s − 5·29-s + 9·31-s + 2·33-s + 35-s − 2·41-s + 43-s − 45-s + 3·47-s + 49-s − 9·53-s + 2·55-s − 57-s − 2·61-s − 63-s + 10·67-s + 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s + 0.229·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s − 0.928·29-s + 1.61·31-s + 0.348·33-s + 0.169·35-s − 0.312·41-s + 0.152·43-s − 0.149·45-s + 0.437·47-s + 1/7·49-s − 1.23·53-s + 0.269·55-s − 0.132·57-s − 0.256·61-s − 0.125·63-s + 1.22·67-s + 0.361·69-s + ⋯

Functional equation

Λ(s)=(56784s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(56784s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5678456784    =    24⋅3⋅7⋅1322^{4} \cdot 3 \cdot 7 \cdot 13^{2}
Sign: 11
Analytic conductor: 453.422453.422
Root analytic conductor: 21.293721.2937
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 56784, ( :1/2), 1)(2,\ 56784,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.57527742350.5752774235
L(12)L(\frac12) ≈\approx 0.57527742350.5752774235
L(32)L(\frac{3}{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+T 1 + T
7 1+T 1 + T
13 1 1
good5 1+T+pT2 1 + T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1+15T+pT2 1 + 15 T + p T^{2}
79 1+11T+pT2 1 + 11 T + p T^{2}
83 1−3T+pT2 1 - 3 T + p T^{2}
89 1−17T+pT2 1 - 17 T + p T^{2}
97 1+3T+pT2 1 + 3 T + p 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

−14.40034432631267, −13.78857174163669, −13.19845246391241, −12.97957403431803, −12.18382641206145, −11.79254305480230, −11.52216636659775, −10.67656922839172, −10.41190531724601, −9.774621961627515, −9.361470288079016, −8.614794854576367, −7.981185113154679, −7.634493184417992, −7.029173697280145, −6.361415992278516, −5.894247094948070, −5.375222349809926, −4.636092009923224, −4.182686756512651, −3.479155125008865, −2.845682596435437, −2.081527457379423, −1.248979155476029, −0.2835377398057606, 0.2835377398057606, 1.248979155476029, 2.081527457379423, 2.845682596435437, 3.479155125008865, 4.182686756512651, 4.636092009923224, 5.375222349809926, 5.894247094948070, 6.361415992278516, 7.029173697280145, 7.634493184417992, 7.981185113154679, 8.614794854576367, 9.361470288079016, 9.774621961627515, 10.41190531724601, 10.67656922839172, 11.52216636659775, 11.79254305480230, 12.18382641206145, 12.97957403431803, 13.19845246391241, 13.78857174163669, 14.40034432631267

Graph of the ZZ-function along the critical line