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Properties

Label 2-208-1.1-c9-0-21
Degree 22
Conductor 208208
Sign −1-1
Analytic cond. 107.127107.127
Root an. cond. 10.350210.3502
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 194.·3-s − 920.·5-s − 5.35e3·7-s + 1.80e4·9-s − 7.92e4·11-s + 2.85e4·13-s + 1.78e5·15-s + 4.52e5·17-s − 2.12e5·19-s + 1.04e6·21-s + 7.59e5·23-s − 1.10e6·25-s + 3.15e5·27-s − 9.00e5·29-s − 2.27e6·31-s + 1.54e7·33-s + 4.93e6·35-s − 4.70e6·37-s − 5.54e6·39-s + 3.39e7·41-s + 2.33e7·43-s − 1.66e7·45-s + 5.14e7·47-s − 1.16e7·49-s − 8.78e7·51-s + 1.01e8·53-s + 7.29e7·55-s + ⋯
L(s)  = 1  − 1.38·3-s − 0.658·5-s − 0.843·7-s + 0.917·9-s − 1.63·11-s + 0.277·13-s + 0.911·15-s + 1.31·17-s − 0.374·19-s + 1.16·21-s + 0.565·23-s − 0.566·25-s + 0.114·27-s − 0.236·29-s − 0.441·31-s + 2.26·33-s + 0.555·35-s − 0.412·37-s − 0.384·39-s + 1.87·41-s + 1.04·43-s − 0.604·45-s + 1.53·47-s − 0.288·49-s − 1.81·51-s + 1.75·53-s + 1.07·55-s + ⋯

Functional equation

Λ(s)=(208s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(208s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 208208    =    24⋅132^{4} \cdot 13
Sign: −1-1
Analytic conductor: 107.127107.127
Root analytic conductor: 10.350210.3502
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 208, ( :9/2), −1)(2,\ 208,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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
13 1−2.85e4T 1 - 2.85e4T
good3 1+194.T+1.96e4T2 1 + 194.T + 1.96e4T^{2}
5 1+920.T+1.95e6T2 1 + 920.T + 1.95e6T^{2}
7 1+5.35e3T+4.03e7T2 1 + 5.35e3T + 4.03e7T^{2}
11 1+7.92e4T+2.35e9T2 1 + 7.92e4T + 2.35e9T^{2}
17 1−4.52e5T+1.18e11T2 1 - 4.52e5T + 1.18e11T^{2}
19 1+2.12e5T+3.22e11T2 1 + 2.12e5T + 3.22e11T^{2}
23 1−7.59e5T+1.80e12T2 1 - 7.59e5T + 1.80e12T^{2}
29 1+9.00e5T+1.45e13T2 1 + 9.00e5T + 1.45e13T^{2}
31 1+2.27e6T+2.64e13T2 1 + 2.27e6T + 2.64e13T^{2}
37 1+4.70e6T+1.29e14T2 1 + 4.70e6T + 1.29e14T^{2}
41 1−3.39e7T+3.27e14T2 1 - 3.39e7T + 3.27e14T^{2}
43 1−2.33e7T+5.02e14T2 1 - 2.33e7T + 5.02e14T^{2}
47 1−5.14e7T+1.11e15T2 1 - 5.14e7T + 1.11e15T^{2}
53 1−1.01e8T+3.29e15T2 1 - 1.01e8T + 3.29e15T^{2}
59 1+1.32e8T+8.66e15T2 1 + 1.32e8T + 8.66e15T^{2}
61 1+1.23e8T+1.16e16T2 1 + 1.23e8T + 1.16e16T^{2}
67 1−2.15e8T+2.72e16T2 1 - 2.15e8T + 2.72e16T^{2}
71 1−2.06e8T+4.58e16T2 1 - 2.06e8T + 4.58e16T^{2}
73 1−3.44e8T+5.88e16T2 1 - 3.44e8T + 5.88e16T^{2}
79 1+5.03e7T+1.19e17T2 1 + 5.03e7T + 1.19e17T^{2}
83 1+8.20e7T+1.86e17T2 1 + 8.20e7T + 1.86e17T^{2}
89 1+6.17e8T+3.50e17T2 1 + 6.17e8T + 3.50e17T^{2}
97 1+9.91e8T+7.60e17T2 1 + 9.91e8T + 7.60e17T^{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

−10.60355406415164528432230989568, −9.516816150832771484575049687336, −8.011033037734526426908897214292, −7.16455700611799142717693917448, −5.90425844863114287575812582517, −5.33480214382167014845621576701, −3.99912029286932311295399467493, −2.72932067970497987323170898050, −0.816540532126358517243907746039, 0, 0.816540532126358517243907746039, 2.72932067970497987323170898050, 3.99912029286932311295399467493, 5.33480214382167014845621576701, 5.90425844863114287575812582517, 7.16455700611799142717693917448, 8.011033037734526426908897214292, 9.516816150832771484575049687336, 10.60355406415164528432230989568

Graph of the ZZ-function along the critical line