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Properties

Label 2-13e2-1.1-c9-0-41
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  − 6.13·2-s − 213.·3-s − 474.·4-s + 517.·5-s + 1.30e3·6-s + 2.96e3·7-s + 6.05e3·8-s + 2.58e4·9-s − 3.17e3·10-s + 9.53e4·11-s + 1.01e5·12-s − 1.81e4·14-s − 1.10e5·15-s + 2.05e5·16-s + 4.44e5·17-s − 1.58e5·18-s + 2.31e5·19-s − 2.45e5·20-s − 6.31e5·21-s − 5.85e5·22-s + 5.50e5·23-s − 1.29e6·24-s − 1.68e6·25-s − 1.31e6·27-s − 1.40e6·28-s + 5.22e6·29-s + 6.78e5·30-s + ⋯
L(s)  = 1  − 0.271·2-s − 1.52·3-s − 0.926·4-s + 0.370·5-s + 0.412·6-s + 0.466·7-s + 0.522·8-s + 1.31·9-s − 0.100·10-s + 1.96·11-s + 1.40·12-s − 0.126·14-s − 0.563·15-s + 0.784·16-s + 1.29·17-s − 0.355·18-s + 0.407·19-s − 0.343·20-s − 0.708·21-s − 0.532·22-s + 0.410·23-s − 0.794·24-s − 0.862·25-s − 0.474·27-s − 0.431·28-s + 1.37·29-s + 0.152·30-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 1.2835956041.283595604
L(12)L(\frac12) ≈\approx 1.2835956041.283595604
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+6.13T+512T2 1 + 6.13T + 512T^{2}
3 1+213.T+1.96e4T2 1 + 213.T + 1.96e4T^{2}
5 1−517.T+1.95e6T2 1 - 517.T + 1.95e6T^{2}
7 1−2.96e3T+4.03e7T2 1 - 2.96e3T + 4.03e7T^{2}
11 1−9.53e4T+2.35e9T2 1 - 9.53e4T + 2.35e9T^{2}
17 1−4.44e5T+1.18e11T2 1 - 4.44e5T + 1.18e11T^{2}
19 1−2.31e5T+3.22e11T2 1 - 2.31e5T + 3.22e11T^{2}
23 1−5.50e5T+1.80e12T2 1 - 5.50e5T + 1.80e12T^{2}
29 1−5.22e6T+1.45e13T2 1 - 5.22e6T + 1.45e13T^{2}
31 1+3.37e6T+2.64e13T2 1 + 3.37e6T + 2.64e13T^{2}
37 1−1.68e7T+1.29e14T2 1 - 1.68e7T + 1.29e14T^{2}
41 1+9.64e6T+3.27e14T2 1 + 9.64e6T + 3.27e14T^{2}
43 1−2.65e7T+5.02e14T2 1 - 2.65e7T + 5.02e14T^{2}
47 1+3.77e7T+1.11e15T2 1 + 3.77e7T + 1.11e15T^{2}
53 1+5.46e6T+3.29e15T2 1 + 5.46e6T + 3.29e15T^{2}
59 1+3.79e7T+8.66e15T2 1 + 3.79e7T + 8.66e15T^{2}
61 1−2.04e8T+1.16e16T2 1 - 2.04e8T + 1.16e16T^{2}
67 1−1.25e8T+2.72e16T2 1 - 1.25e8T + 2.72e16T^{2}
71 1−2.01e8T+4.58e16T2 1 - 2.01e8T + 4.58e16T^{2}
73 1−3.18e8T+5.88e16T2 1 - 3.18e8T + 5.88e16T^{2}
79 1+5.08e8T+1.19e17T2 1 + 5.08e8T + 1.19e17T^{2}
83 1−3.82e8T+1.86e17T2 1 - 3.82e8T + 1.86e17T^{2}
89 1−2.76e8T+3.50e17T2 1 - 2.76e8T + 3.50e17T^{2}
97 1+1.36e9T+7.60e17T2 1 + 1.36e9T + 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

−11.21944370725805284325549700626, −9.995297752754083524528117554520, −9.371782687811635471114830170877, −8.055087794542953506233713147965, −6.68753392595907144983853351407, −5.73618634039603345145101132483, −4.84137740180275210801943312443, −3.79859975770035291243390334172, −1.34532571698420929752655415668, −0.77112213380049960808445528132, 0.77112213380049960808445528132, 1.34532571698420929752655415668, 3.79859975770035291243390334172, 4.84137740180275210801943312443, 5.73618634039603345145101132483, 6.68753392595907144983853351407, 8.055087794542953506233713147965, 9.371782687811635471114830170877, 9.995297752754083524528117554520, 11.21944370725805284325549700626

Graph of the ZZ-function along the critical line