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Properties

Label 2-45-1.1-c11-0-4
Degree 22
Conductor 4545
Sign 11
Analytic cond. 34.575434.5754
Root an. cond. 5.880085.88008
Motivic weight 1111
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.7·2-s − 1.83e3·4-s + 3.12e3·5-s + 7.99e4·7-s + 5.71e4·8-s − 4.59e4·10-s − 8.05e5·11-s − 1.19e6·13-s − 1.17e6·14-s + 2.91e6·16-s − 2.63e6·17-s + 1.16e7·19-s − 5.72e6·20-s + 1.18e7·22-s − 1.84e7·23-s + 9.76e6·25-s + 1.76e7·26-s − 1.46e8·28-s + 1.90e8·29-s + 1.01e8·31-s − 1.59e8·32-s + 3.87e7·34-s + 2.49e8·35-s + 8.06e7·37-s − 1.70e8·38-s + 1.78e8·40-s − 2.26e8·41-s + ⋯
L(s)  = 1  − 0.325·2-s − 0.894·4-s + 0.447·5-s + 1.79·7-s + 0.616·8-s − 0.145·10-s − 1.50·11-s − 0.894·13-s − 0.584·14-s + 0.693·16-s − 0.449·17-s + 1.07·19-s − 0.399·20-s + 0.490·22-s − 0.596·23-s + 0.199·25-s + 0.290·26-s − 1.60·28-s + 1.72·29-s + 0.634·31-s − 0.841·32-s + 0.146·34-s + 0.803·35-s + 0.191·37-s − 0.349·38-s + 0.275·40-s − 0.305·41-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=( Λ(12−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+11/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 11
Analytic conductor: 34.575434.5754
Root analytic conductor: 5.880085.88008
Motivic weight: 1111
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 45, ( :11/2), 1)(2,\ 45,\ (\ :11/2),\ 1)

Particular Values

L(6)L(6) ≈\approx 1.6016917751.601691775
L(12)L(\frac12) ≈\approx 1.6016917751.601691775
L(132)L(\frac{13}{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 1−3.12e3T 1 - 3.12e3T
good2 1+14.7T+2.04e3T2 1 + 14.7T + 2.04e3T^{2}
7 1−7.99e4T+1.97e9T2 1 - 7.99e4T + 1.97e9T^{2}
11 1+8.05e5T+2.85e11T2 1 + 8.05e5T + 2.85e11T^{2}
13 1+1.19e6T+1.79e12T2 1 + 1.19e6T + 1.79e12T^{2}
17 1+2.63e6T+3.42e13T2 1 + 2.63e6T + 3.42e13T^{2}
19 1−1.16e7T+1.16e14T2 1 - 1.16e7T + 1.16e14T^{2}
23 1+1.84e7T+9.52e14T2 1 + 1.84e7T + 9.52e14T^{2}
29 1−1.90e8T+1.22e16T2 1 - 1.90e8T + 1.22e16T^{2}
31 1−1.01e8T+2.54e16T2 1 - 1.01e8T + 2.54e16T^{2}
37 1−8.06e7T+1.77e17T2 1 - 8.06e7T + 1.77e17T^{2}
41 1+2.26e8T+5.50e17T2 1 + 2.26e8T + 5.50e17T^{2}
43 1−1.67e9T+9.29e17T2 1 - 1.67e9T + 9.29e17T^{2}
47 1+8.58e8T+2.47e18T2 1 + 8.58e8T + 2.47e18T^{2}
53 1−3.52e9T+9.26e18T2 1 - 3.52e9T + 9.26e18T^{2}
59 1+4.35e9T+3.01e19T2 1 + 4.35e9T + 3.01e19T^{2}
61 1+1.65e9T+4.35e19T2 1 + 1.65e9T + 4.35e19T^{2}
67 1−7.58e9T+1.22e20T2 1 - 7.58e9T + 1.22e20T^{2}
71 1−2.75e10T+2.31e20T2 1 - 2.75e10T + 2.31e20T^{2}
73 1−3.22e10T+3.13e20T2 1 - 3.22e10T + 3.13e20T^{2}
79 1+2.43e9T+7.47e20T2 1 + 2.43e9T + 7.47e20T^{2}
83 1+1.20e10T+1.28e21T2 1 + 1.20e10T + 1.28e21T^{2}
89 1+4.44e9T+2.77e21T2 1 + 4.44e9T + 2.77e21T^{2}
97 1+2.04e10T+7.15e21T2 1 + 2.04e10T + 7.15e21T^{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

−13.68193435754841657255604354637, −12.26406802831175901985188697756, −10.81561993512749481374342606512, −9.824922830464044406537568554503, −8.380038696439269686947720158145, −7.64906595273599130507597497486, −5.30302606493196631508172375860, −4.64352600685496654204004490722, −2.34645732011858897521570874701, −0.839303104314245879140973879268, 0.839303104314245879140973879268, 2.34645732011858897521570874701, 4.64352600685496654204004490722, 5.30302606493196631508172375860, 7.64906595273599130507597497486, 8.380038696439269686947720158145, 9.824922830464044406537568554503, 10.81561993512749481374342606512, 12.26406802831175901985188697756, 13.68193435754841657255604354637

Graph of the ZZ-function along the critical line