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Properties

Label 2-13e2-1.1-c9-0-10
Degree 22
Conductor 169169
Sign 11
Analytic cond. 87.041087.0410
Root an. cond. 9.329579.32957
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  + 13.4·2-s + 117.·3-s − 331.·4-s − 2.29e3·5-s + 1.58e3·6-s − 5.05e3·7-s − 1.13e4·8-s − 5.81e3·9-s − 3.07e4·10-s − 8.13e3·11-s − 3.90e4·12-s − 6.79e4·14-s − 2.69e5·15-s + 1.73e4·16-s − 4.81e5·17-s − 7.81e4·18-s + 5.84e5·19-s + 7.59e5·20-s − 5.95e5·21-s − 1.09e5·22-s − 2.43e6·23-s − 1.33e6·24-s + 3.29e6·25-s − 3.00e6·27-s + 1.67e6·28-s − 3.48e6·29-s − 3.62e6·30-s + ⋯
L(s)  = 1  + 0.593·2-s + 0.839·3-s − 0.647·4-s − 1.63·5-s + 0.498·6-s − 0.796·7-s − 0.978·8-s − 0.295·9-s − 0.973·10-s − 0.167·11-s − 0.543·12-s − 0.473·14-s − 1.37·15-s + 0.0660·16-s − 1.39·17-s − 0.175·18-s + 1.02·19-s + 1.06·20-s − 0.668·21-s − 0.0995·22-s − 1.81·23-s − 0.821·24-s + 1.68·25-s − 1.08·27-s + 0.515·28-s − 0.916·29-s − 0.817·30-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 11
Analytic conductor: 87.041087.0410
Root analytic conductor: 9.329579.32957
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 169, ( :9/2), 1)(2,\ 169,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 0.55174420020.5517442002
L(12)L(\frac12) ≈\approx 0.55174420020.5517442002
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)
bad13 1 1
good2 1−13.4T+512T2 1 - 13.4T + 512T^{2}
3 1−117.T+1.96e4T2 1 - 117.T + 1.96e4T^{2}
5 1+2.29e3T+1.95e6T2 1 + 2.29e3T + 1.95e6T^{2}
7 1+5.05e3T+4.03e7T2 1 + 5.05e3T + 4.03e7T^{2}
11 1+8.13e3T+2.35e9T2 1 + 8.13e3T + 2.35e9T^{2}
17 1+4.81e5T+1.18e11T2 1 + 4.81e5T + 1.18e11T^{2}
19 1−5.84e5T+3.22e11T2 1 - 5.84e5T + 3.22e11T^{2}
23 1+2.43e6T+1.80e12T2 1 + 2.43e6T + 1.80e12T^{2}
29 1+3.48e6T+1.45e13T2 1 + 3.48e6T + 1.45e13T^{2}
31 1−2.77e6T+2.64e13T2 1 - 2.77e6T + 2.64e13T^{2}
37 1+8.63e6T+1.29e14T2 1 + 8.63e6T + 1.29e14T^{2}
41 1−4.42e6T+3.27e14T2 1 - 4.42e6T + 3.27e14T^{2}
43 1+1.52e7T+5.02e14T2 1 + 1.52e7T + 5.02e14T^{2}
47 1−5.12e7T+1.11e15T2 1 - 5.12e7T + 1.11e15T^{2}
53 1−9.09e7T+3.29e15T2 1 - 9.09e7T + 3.29e15T^{2}
59 1−1.20e8T+8.66e15T2 1 - 1.20e8T + 8.66e15T^{2}
61 1−1.61e8T+1.16e16T2 1 - 1.61e8T + 1.16e16T^{2}
67 1+1.69e7T+2.72e16T2 1 + 1.69e7T + 2.72e16T^{2}
71 1−3.87e7T+4.58e16T2 1 - 3.87e7T + 4.58e16T^{2}
73 1+3.71e8T+5.88e16T2 1 + 3.71e8T + 5.88e16T^{2}
79 1+2.90e8T+1.19e17T2 1 + 2.90e8T + 1.19e17T^{2}
83 1+6.61e7T+1.86e17T2 1 + 6.61e7T + 1.86e17T^{2}
89 1+5.15e8T+3.50e17T2 1 + 5.15e8T + 3.50e17T^{2}
97 1+1.21e9T+7.60e17T2 1 + 1.21e9T + 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

−11.45515797421618382168636858897, −9.901314402388447800410448349649, −8.819569574186548960052526734845, −8.204419687119722590493524350675, −7.09097034028184164890935309543, −5.62002697159388314009194207326, −4.14995652308792379693490378153, −3.66595135220253799211062890263, −2.63054842292787124980352537159, −0.30539367595966465353521831565, 0.30539367595966465353521831565, 2.63054842292787124980352537159, 3.66595135220253799211062890263, 4.14995652308792379693490378153, 5.62002697159388314009194207326, 7.09097034028184164890935309543, 8.204419687119722590493524350675, 8.819569574186548960052526734845, 9.901314402388447800410448349649, 11.45515797421618382168636858897

Graph of the ZZ-function along the critical line