Properties

Label 2-57-1.1-c9-0-13
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  + 26.5·2-s + 81·3-s + 190.·4-s + 1.29e3·5-s + 2.14e3·6-s − 3.38e3·7-s − 8.52e3·8-s + 6.56e3·9-s + 3.43e4·10-s + 8.29e4·11-s + 1.54e4·12-s + 1.56e5·13-s − 8.97e4·14-s + 1.05e5·15-s − 3.23e5·16-s + 1.22e5·17-s + 1.73e5·18-s − 1.30e5·19-s + 2.47e5·20-s − 2.74e5·21-s + 2.19e6·22-s + 1.72e6·23-s − 6.90e5·24-s − 2.71e5·25-s + 4.13e6·26-s + 5.31e5·27-s − 6.44e5·28-s + ⋯
L(s)  = 1  + 1.17·2-s + 0.577·3-s + 0.372·4-s + 0.928·5-s + 0.676·6-s − 0.532·7-s − 0.735·8-s + 0.333·9-s + 1.08·10-s + 1.70·11-s + 0.214·12-s + 1.51·13-s − 0.624·14-s + 0.535·15-s − 1.23·16-s + 0.355·17-s + 0.390·18-s − 0.229·19-s + 0.345·20-s − 0.307·21-s + 2.00·22-s + 1.28·23-s − 0.424·24-s − 0.138·25-s + 1.77·26-s + 0.192·27-s − 0.198·28-s + ⋯

Functional equation

Λ(s)=(57s/2ΓC(s)L(s)=(Λ(10s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(57s/2ΓC(s+9/2)L(s)=(Λ(1s)\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    =    3193 \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 5.1445758765.144575876
L(12)L(\frac12) \approx 5.1445758765.144575876
L(112)L(\frac{11}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 181T 1 - 81T
19 1+1.30e5T 1 + 1.30e5T
good2 126.5T+512T2 1 - 26.5T + 512T^{2}
5 11.29e3T+1.95e6T2 1 - 1.29e3T + 1.95e6T^{2}
7 1+3.38e3T+4.03e7T2 1 + 3.38e3T + 4.03e7T^{2}
11 18.29e4T+2.35e9T2 1 - 8.29e4T + 2.35e9T^{2}
13 11.56e5T+1.06e10T2 1 - 1.56e5T + 1.06e10T^{2}
17 11.22e5T+1.18e11T2 1 - 1.22e5T + 1.18e11T^{2}
23 11.72e6T+1.80e12T2 1 - 1.72e6T + 1.80e12T^{2}
29 1+1.25e5T+1.45e13T2 1 + 1.25e5T + 1.45e13T^{2}
31 1+2.93e6T+2.64e13T2 1 + 2.93e6T + 2.64e13T^{2}
37 16.00e5T+1.29e14T2 1 - 6.00e5T + 1.29e14T^{2}
41 12.80e7T+3.27e14T2 1 - 2.80e7T + 3.27e14T^{2}
43 1+1.16e7T+5.02e14T2 1 + 1.16e7T + 5.02e14T^{2}
47 1+8.08e6T+1.11e15T2 1 + 8.08e6T + 1.11e15T^{2}
53 1+2.67e7T+3.29e15T2 1 + 2.67e7T + 3.29e15T^{2}
59 19.71e7T+8.66e15T2 1 - 9.71e7T + 8.66e15T^{2}
61 1+1.59e8T+1.16e16T2 1 + 1.59e8T + 1.16e16T^{2}
67 1+2.71e8T+2.72e16T2 1 + 2.71e8T + 2.72e16T^{2}
71 13.38e8T+4.58e16T2 1 - 3.38e8T + 4.58e16T^{2}
73 1+7.79e7T+5.88e16T2 1 + 7.79e7T + 5.88e16T^{2}
79 1+5.96e8T+1.19e17T2 1 + 5.96e8T + 1.19e17T^{2}
83 1+7.10e7T+1.86e17T2 1 + 7.10e7T + 1.86e17T^{2}
89 1+1.04e9T+3.50e17T2 1 + 1.04e9T + 3.50e17T^{2}
97 13.58e8T+7.60e17T2 1 - 3.58e8T + 7.60e17T^{2}
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   L(s)=p j=12(1αj,pps)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

−13.44147254156790786017096318296, −12.63519953212839916704914774145, −11.25375130056020728367951185724, −9.529606262605644189790777064501, −8.837884647736523485960891868210, −6.64361359829163123598492083350, −5.82301459918773624724813704423, −4.12659123642001577064196468572, −3.13822252147872819712721812270, −1.40469625205078531351679646561, 1.40469625205078531351679646561, 3.13822252147872819712721812270, 4.12659123642001577064196468572, 5.82301459918773624724813704423, 6.64361359829163123598492083350, 8.837884647736523485960891868210, 9.529606262605644189790777064501, 11.25375130056020728367951185724, 12.63519953212839916704914774145, 13.44147254156790786017096318296

Graph of the ZZ-function along the critical line