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Properties

Label 2-16245-1.1-c1-0-4
Degree 22
Conductor 1624516245
Sign −1-1
Analytic cond. 129.716129.716
Root an. cond. 11.389311.3893
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s − 4·7-s − 3·11-s + 2·13-s + 4·16-s − 6·17-s − 2·20-s + 25-s + 8·28-s + 3·29-s − 7·31-s − 4·35-s + 8·37-s + 6·41-s − 4·43-s + 6·44-s − 6·47-s + 9·49-s − 4·52-s + 6·53-s − 3·55-s + 15·59-s + 5·61-s − 8·64-s + 2·65-s + 2·67-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 1.51·7-s − 0.904·11-s + 0.554·13-s + 16-s − 1.45·17-s − 0.447·20-s + 1/5·25-s + 1.51·28-s + 0.557·29-s − 1.25·31-s − 0.676·35-s + 1.31·37-s + 0.937·41-s − 0.609·43-s + 0.904·44-s − 0.875·47-s + 9/7·49-s − 0.554·52-s + 0.824·53-s − 0.404·55-s + 1.95·59-s + 0.640·61-s − 64-s + 0.248·65-s + 0.244·67-s + ⋯

Functional equation

Λ(s)=(16245s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(16245s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1624516245    =    32⋅5⋅1923^{2} \cdot 5 \cdot 19^{2}
Sign: −1-1
Analytic conductor: 129.716129.716
Root analytic conductor: 11.389311.3893
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 16245, ( :1/2), −1)(2,\ 16245,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
5 1−T 1 - T
19 1 1
good2 1+pT2 1 + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−15T+pT2 1 - 15 T + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−15T+pT2 1 - 15 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−16.30119689098478, −15.76343004709598, −15.10875206008760, −14.53473324374217, −13.80838973296866, −13.25813792310624, −12.98045060402339, −12.79172747321526, −11.82154761588390, −10.99027232124414, −10.53614897958416, −9.750125168316176, −9.585535982538966, −8.840256661932381, −8.421781329169140, −7.628873009475114, −6.771058136690333, −6.342456284513648, −5.606872878582650, −5.091029581597595, −4.190628396962405, −3.684027013863426, −2.851846828219073, −2.187955992227127, −0.8353148224984355, 0, 0.8353148224984355, 2.187955992227127, 2.851846828219073, 3.684027013863426, 4.190628396962405, 5.091029581597595, 5.606872878582650, 6.342456284513648, 6.771058136690333, 7.628873009475114, 8.421781329169140, 8.840256661932381, 9.585535982538966, 9.750125168316176, 10.53614897958416, 10.99027232124414, 11.82154761588390, 12.79172747321526, 12.98045060402339, 13.25813792310624, 13.80838973296866, 14.53473324374217, 15.10875206008760, 15.76343004709598, 16.30119689098478

Graph of the ZZ-function along the critical line