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Properties

Label 2-57-1.1-c9-0-7
Degree 22
Conductor 5757
Sign 11
Analytic cond. 29.357029.3570
Root an. cond. 5.418215.41821
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  − 18.3·2-s − 81·3-s − 173.·4-s + 1.62e3·5-s + 1.48e3·6-s + 7.42e3·7-s + 1.26e4·8-s + 6.56e3·9-s − 2.98e4·10-s − 7.47e4·11-s + 1.40e4·12-s − 3.15e4·13-s − 1.36e5·14-s − 1.31e5·15-s − 1.43e5·16-s + 2.75e5·17-s − 1.20e5·18-s + 1.30e5·19-s − 2.82e5·20-s − 6.01e5·21-s + 1.37e6·22-s − 1.03e6·23-s − 1.02e6·24-s + 6.86e5·25-s + 5.79e5·26-s − 5.31e5·27-s − 1.29e6·28-s + ⋯
L(s)  = 1  − 0.812·2-s − 0.577·3-s − 0.339·4-s + 1.16·5-s + 0.469·6-s + 1.16·7-s + 1.08·8-s + 0.333·9-s − 0.944·10-s − 1.53·11-s + 0.195·12-s − 0.306·13-s − 0.950·14-s − 0.671·15-s − 0.545·16-s + 0.799·17-s − 0.270·18-s + 0.229·19-s − 0.394·20-s − 0.674·21-s + 1.25·22-s − 0.771·23-s − 0.628·24-s + 0.351·25-s + 0.248·26-s − 0.192·27-s − 0.396·28-s + ⋯

Functional equation

Λ(s)=(57s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(57s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5757    =    3⋅193 \cdot 19
Sign: 11
Analytic conductor: 29.357029.3570
Root analytic conductor: 5.418215.41821
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 57, ( :9/2), 1)(2,\ 57,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 1.1769335951.176933595
L(12)L(\frac12) ≈\approx 1.1769335951.176933595
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+81T 1 + 81T
19 1−1.30e5T 1 - 1.30e5T
good2 1+18.3T+512T2 1 + 18.3T + 512T^{2}
5 1−1.62e3T+1.95e6T2 1 - 1.62e3T + 1.95e6T^{2}
7 1−7.42e3T+4.03e7T2 1 - 7.42e3T + 4.03e7T^{2}
11 1+7.47e4T+2.35e9T2 1 + 7.47e4T + 2.35e9T^{2}
13 1+3.15e4T+1.06e10T2 1 + 3.15e4T + 1.06e10T^{2}
17 1−2.75e5T+1.18e11T2 1 - 2.75e5T + 1.18e11T^{2}
23 1+1.03e6T+1.80e12T2 1 + 1.03e6T + 1.80e12T^{2}
29 1−3.07e6T+1.45e13T2 1 - 3.07e6T + 1.45e13T^{2}
31 1−9.42e6T+2.64e13T2 1 - 9.42e6T + 2.64e13T^{2}
37 1+1.67e7T+1.29e14T2 1 + 1.67e7T + 1.29e14T^{2}
41 1+1.58e7T+3.27e14T2 1 + 1.58e7T + 3.27e14T^{2}
43 1−2.72e7T+5.02e14T2 1 - 2.72e7T + 5.02e14T^{2}
47 1−4.46e7T+1.11e15T2 1 - 4.46e7T + 1.11e15T^{2}
53 1−9.15e7T+3.29e15T2 1 - 9.15e7T + 3.29e15T^{2}
59 1−1.67e8T+8.66e15T2 1 - 1.67e8T + 8.66e15T^{2}
61 1+5.39e7T+1.16e16T2 1 + 5.39e7T + 1.16e16T^{2}
67 1+2.55e8T+2.72e16T2 1 + 2.55e8T + 2.72e16T^{2}
71 1−2.56e8T+4.58e16T2 1 - 2.56e8T + 4.58e16T^{2}
73 1+2.80e8T+5.88e16T2 1 + 2.80e8T + 5.88e16T^{2}
79 1−1.85e8T+1.19e17T2 1 - 1.85e8T + 1.19e17T^{2}
83 1−1.79e8T+1.86e17T2 1 - 1.79e8T + 1.86e17T^{2}
89 1−1.28e8T+3.50e17T2 1 - 1.28e8T + 3.50e17T^{2}
97 1−9.23e8T+7.60e17T2 1 - 9.23e8T + 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

−13.45098227953455890143237215494, −12.01604450528439635890335197297, −10.40071962435550651390092212582, −10.11742513034691025141163551454, −8.536813422571917357286852728546, −7.53254064425732765038100810834, −5.62243600178029516227108758436, −4.78411227514485472029163337560, −2.12959690721544624881011367216, −0.826245364016455317476660575095, 0.826245364016455317476660575095, 2.12959690721544624881011367216, 4.78411227514485472029163337560, 5.62243600178029516227108758436, 7.53254064425732765038100810834, 8.536813422571917357286852728546, 10.11742513034691025141163551454, 10.40071962435550651390092212582, 12.01604450528439635890335197297, 13.45098227953455890143237215494

Graph of the ZZ-function along the critical line