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Properties

Label 2-315-1.1-c9-0-9
Degree 22
Conductor 315315
Sign 11
Analytic cond. 162.236162.236
Root an. cond. 12.737212.7372
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.05·2-s − 447.·4-s − 625·5-s − 2.40e3·7-s + 7.72e3·8-s + 5.03e3·10-s + 4.77e4·11-s − 7.12e4·13-s + 1.93e4·14-s + 1.66e5·16-s + 4.33e4·17-s − 4.63e5·19-s + 2.79e5·20-s − 3.84e5·22-s + 5.02e5·23-s + 3.90e5·25-s + 5.73e5·26-s + 1.07e6·28-s + 3.95e6·29-s + 1.81e5·31-s − 5.29e6·32-s − 3.49e5·34-s + 1.50e6·35-s − 1.74e7·37-s + 3.73e6·38-s − 4.82e6·40-s − 3.40e7·41-s + ⋯
L(s)  = 1  − 0.355·2-s − 0.873·4-s − 0.447·5-s − 0.377·7-s + 0.666·8-s + 0.159·10-s + 0.982·11-s − 0.691·13-s + 0.134·14-s + 0.636·16-s + 0.125·17-s − 0.816·19-s + 0.390·20-s − 0.349·22-s + 0.374·23-s + 0.200·25-s + 0.246·26-s + 0.330·28-s + 1.03·29-s + 0.0353·31-s − 0.893·32-s − 0.0448·34-s + 0.169·35-s − 1.53·37-s + 0.290·38-s − 0.298·40-s − 1.88·41-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 162.236162.236
Root analytic conductor: 12.737212.7372
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 315, ( :9/2), 1)(2,\ 315,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 0.77676436840.7767643684
L(12)L(\frac12) ≈\approx 0.77676436840.7767643684
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)
bad3 1 1
5 1+625T 1 + 625T
7 1+2.40e3T 1 + 2.40e3T
good2 1+8.05T+512T2 1 + 8.05T + 512T^{2}
11 1−4.77e4T+2.35e9T2 1 - 4.77e4T + 2.35e9T^{2}
13 1+7.12e4T+1.06e10T2 1 + 7.12e4T + 1.06e10T^{2}
17 1−4.33e4T+1.18e11T2 1 - 4.33e4T + 1.18e11T^{2}
19 1+4.63e5T+3.22e11T2 1 + 4.63e5T + 3.22e11T^{2}
23 1−5.02e5T+1.80e12T2 1 - 5.02e5T + 1.80e12T^{2}
29 1−3.95e6T+1.45e13T2 1 - 3.95e6T + 1.45e13T^{2}
31 1−1.81e5T+2.64e13T2 1 - 1.81e5T + 2.64e13T^{2}
37 1+1.74e7T+1.29e14T2 1 + 1.74e7T + 1.29e14T^{2}
41 1+3.40e7T+3.27e14T2 1 + 3.40e7T + 3.27e14T^{2}
43 1−4.15e7T+5.02e14T2 1 - 4.15e7T + 5.02e14T^{2}
47 1+2.41e6T+1.11e15T2 1 + 2.41e6T + 1.11e15T^{2}
53 1+1.01e8T+3.29e15T2 1 + 1.01e8T + 3.29e15T^{2}
59 1+4.60e7T+8.66e15T2 1 + 4.60e7T + 8.66e15T^{2}
61 1+1.04e8T+1.16e16T2 1 + 1.04e8T + 1.16e16T^{2}
67 1−1.90e8T+2.72e16T2 1 - 1.90e8T + 2.72e16T^{2}
71 1+1.98e8T+4.58e16T2 1 + 1.98e8T + 4.58e16T^{2}
73 1+3.32e8T+5.88e16T2 1 + 3.32e8T + 5.88e16T^{2}
79 1−3.57e8T+1.19e17T2 1 - 3.57e8T + 1.19e17T^{2}
83 1−3.84e7T+1.86e17T2 1 - 3.84e7T + 1.86e17T^{2}
89 1−2.06e7T+3.50e17T2 1 - 2.06e7T + 3.50e17T^{2}
97 1+1.06e8T+7.60e17T2 1 + 1.06e8T + 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

−9.971827047736543396213576789061, −9.099555159852882537313652709777, −8.424584241122893216747432376033, −7.35094885444320571734963509662, −6.37983521700026073231362225020, −5.00688258548392627113249146455, −4.18140030070642521954758674095, −3.16801150812862523205257615068, −1.57709836941387653183156151435, −0.42434586945998333900352262217, 0.42434586945998333900352262217, 1.57709836941387653183156151435, 3.16801150812862523205257615068, 4.18140030070642521954758674095, 5.00688258548392627113249146455, 6.37983521700026073231362225020, 7.35094885444320571734963509662, 8.424584241122893216747432376033, 9.099555159852882537313652709777, 9.971827047736543396213576789061

Graph of the ZZ-function along the critical line