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Properties

Label 2-13-1.1-c9-0-8
Degree 22
Conductor 1313
Sign −1-1
Analytic cond. 6.695466.69546
Root an. cond. 2.587552.58755
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 28.8·2-s − 204.·3-s + 317.·4-s − 258.·5-s − 5.89e3·6-s − 8.86e3·7-s − 5.59e3·8-s + 2.21e4·9-s − 7.45e3·10-s + 3.60e4·11-s − 6.49e4·12-s − 2.85e4·13-s − 2.55e5·14-s + 5.29e4·15-s − 3.23e5·16-s + 3.27e5·17-s + 6.38e5·18-s + 2.65e5·19-s − 8.22e4·20-s + 1.81e6·21-s + 1.03e6·22-s − 2.42e6·23-s + 1.14e6·24-s − 1.88e6·25-s − 8.22e5·26-s − 5.09e5·27-s − 2.81e6·28-s + ⋯
L(s)  = 1  + 1.27·2-s − 1.45·3-s + 0.620·4-s − 0.185·5-s − 1.85·6-s − 1.39·7-s − 0.483·8-s + 1.12·9-s − 0.235·10-s + 0.743·11-s − 0.904·12-s − 0.277·13-s − 1.77·14-s + 0.270·15-s − 1.23·16-s + 0.950·17-s + 1.43·18-s + 0.467·19-s − 0.114·20-s + 2.03·21-s + 0.946·22-s − 1.80·23-s + 0.704·24-s − 0.965·25-s − 0.353·26-s − 0.184·27-s − 0.865·28-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: −1-1
Analytic conductor: 6.695466.69546
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 13, ( :9/2), −1)(2,\ 13,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
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+2.85e4T 1 + 2.85e4T
good2 1−28.8T+512T2 1 - 28.8T + 512T^{2}
3 1+204.T+1.96e4T2 1 + 204.T + 1.96e4T^{2}
5 1+258.T+1.95e6T2 1 + 258.T + 1.95e6T^{2}
7 1+8.86e3T+4.03e7T2 1 + 8.86e3T + 4.03e7T^{2}
11 1−3.60e4T+2.35e9T2 1 - 3.60e4T + 2.35e9T^{2}
17 1−3.27e5T+1.18e11T2 1 - 3.27e5T + 1.18e11T^{2}
19 1−2.65e5T+3.22e11T2 1 - 2.65e5T + 3.22e11T^{2}
23 1+2.42e6T+1.80e12T2 1 + 2.42e6T + 1.80e12T^{2}
29 1−3.99e6T+1.45e13T2 1 - 3.99e6T + 1.45e13T^{2}
31 1+6.45e6T+2.64e13T2 1 + 6.45e6T + 2.64e13T^{2}
37 1+8.15e6T+1.29e14T2 1 + 8.15e6T + 1.29e14T^{2}
41 1+7.20e5T+3.27e14T2 1 + 7.20e5T + 3.27e14T^{2}
43 1+4.13e7T+5.02e14T2 1 + 4.13e7T + 5.02e14T^{2}
47 1−3.67e7T+1.11e15T2 1 - 3.67e7T + 1.11e15T^{2}
53 1−1.67e7T+3.29e15T2 1 - 1.67e7T + 3.29e15T^{2}
59 1+5.89e7T+8.66e15T2 1 + 5.89e7T + 8.66e15T^{2}
61 1−1.28e8T+1.16e16T2 1 - 1.28e8T + 1.16e16T^{2}
67 1+1.91e8T+2.72e16T2 1 + 1.91e8T + 2.72e16T^{2}
71 1−3.40e8T+4.58e16T2 1 - 3.40e8T + 4.58e16T^{2}
73 1−3.19e8T+5.88e16T2 1 - 3.19e8T + 5.88e16T^{2}
79 1−2.44e8T+1.19e17T2 1 - 2.44e8T + 1.19e17T^{2}
83 1+4.38e8T+1.86e17T2 1 + 4.38e8T + 1.86e17T^{2}
89 1+7.90e8T+3.50e17T2 1 + 7.90e8T + 3.50e17T^{2}
97 1+1.65e8T+7.60e17T2 1 + 1.65e8T + 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

−16.69703621810807532671241614519, −15.72781718784338550861644124803, −13.95928460430890317870092855744, −12.43626443260267289556638154263, −11.83918470565123820424209471048, −9.892948414715274431333698709278, −6.57639634467340707158950518251, −5.54522832179349602028947311014, −3.73592644965051715793501526344, 0, 3.73592644965051715793501526344, 5.54522832179349602028947311014, 6.57639634467340707158950518251, 9.892948414715274431333698709278, 11.83918470565123820424209471048, 12.43626443260267289556638154263, 13.95928460430890317870092855744, 15.72781718784338550861644124803, 16.69703621810807532671241614519

Graph of the ZZ-function along the critical line