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Properties

Label 2-13e2-1.1-c9-0-15
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  + 11.1·2-s − 140.·3-s − 388.·4-s − 992.·5-s − 1.56e3·6-s + 1.05e4·7-s − 1.00e4·8-s + 70.0·9-s − 1.10e4·10-s + 1.31e4·11-s + 5.45e4·12-s + 1.17e5·14-s + 1.39e5·15-s + 8.73e4·16-s − 5.21e5·17-s + 779.·18-s − 9.83e5·19-s + 3.85e5·20-s − 1.48e6·21-s + 1.46e5·22-s − 7.72e5·23-s + 1.40e6·24-s − 9.68e5·25-s + 2.75e6·27-s − 4.10e6·28-s − 5.37e5·29-s + 1.55e6·30-s + ⋯
L(s)  = 1  + 0.491·2-s − 1.00·3-s − 0.758·4-s − 0.709·5-s − 0.492·6-s + 1.66·7-s − 0.864·8-s + 0.00356·9-s − 0.349·10-s + 0.270·11-s + 0.759·12-s + 0.817·14-s + 0.711·15-s + 0.333·16-s − 1.51·17-s + 0.00175·18-s − 1.73·19-s + 0.538·20-s − 1.66·21-s + 0.133·22-s − 0.575·23-s + 0.866·24-s − 0.496·25-s + 0.998·27-s − 1.26·28-s − 0.141·29-s + 0.349·30-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 0.60837952490.6083795249
L(12)L(\frac12) ≈\approx 0.60837952490.6083795249
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 1−11.1T+512T2 1 - 11.1T + 512T^{2}
3 1+140.T+1.96e4T2 1 + 140.T + 1.96e4T^{2}
5 1+992.T+1.95e6T2 1 + 992.T + 1.95e6T^{2}
7 1−1.05e4T+4.03e7T2 1 - 1.05e4T + 4.03e7T^{2}
11 1−1.31e4T+2.35e9T2 1 - 1.31e4T + 2.35e9T^{2}
17 1+5.21e5T+1.18e11T2 1 + 5.21e5T + 1.18e11T^{2}
19 1+9.83e5T+3.22e11T2 1 + 9.83e5T + 3.22e11T^{2}
23 1+7.72e5T+1.80e12T2 1 + 7.72e5T + 1.80e12T^{2}
29 1+5.37e5T+1.45e13T2 1 + 5.37e5T + 1.45e13T^{2}
31 1+1.53e6T+2.64e13T2 1 + 1.53e6T + 2.64e13T^{2}
37 1+1.26e7T+1.29e14T2 1 + 1.26e7T + 1.29e14T^{2}
41 1+2.44e7T+3.27e14T2 1 + 2.44e7T + 3.27e14T^{2}
43 1−1.71e7T+5.02e14T2 1 - 1.71e7T + 5.02e14T^{2}
47 1+3.36e7T+1.11e15T2 1 + 3.36e7T + 1.11e15T^{2}
53 1−4.66e7T+3.29e15T2 1 - 4.66e7T + 3.29e15T^{2}
59 1−7.34e7T+8.66e15T2 1 - 7.34e7T + 8.66e15T^{2}
61 1−1.23e8T+1.16e16T2 1 - 1.23e8T + 1.16e16T^{2}
67 1+1.40e8T+2.72e16T2 1 + 1.40e8T + 2.72e16T^{2}
71 1−2.75e8T+4.58e16T2 1 - 2.75e8T + 4.58e16T^{2}
73 1−1.68e8T+5.88e16T2 1 - 1.68e8T + 5.88e16T^{2}
79 1+5.90e8T+1.19e17T2 1 + 5.90e8T + 1.19e17T^{2}
83 1+2.27e8T+1.86e17T2 1 + 2.27e8T + 1.86e17T^{2}
89 1−7.06e8T+3.50e17T2 1 - 7.06e8T + 3.50e17T^{2}
97 1+7.39e8T+7.60e17T2 1 + 7.39e8T + 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

−11.37115604193396644070987376538, −10.48653766389034196675013671790, −8.736746095232117077096448150918, −8.243345300112972195476067523827, −6.69132096405336939118286685954, −5.50928252472032588270237629808, −4.62959834361245506111026993171, −3.99405031623920950414103457590, −1.97212604537653278504411367083, −0.37274967091435788208787660415, 0.37274967091435788208787660415, 1.97212604537653278504411367083, 3.99405031623920950414103457590, 4.62959834361245506111026993171, 5.50928252472032588270237629808, 6.69132096405336939118286685954, 8.243345300112972195476067523827, 8.736746095232117077096448150918, 10.48653766389034196675013671790, 11.37115604193396644070987376538

Graph of the ZZ-function along the critical line