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Properties

Label 2-13-1.1-c9-0-4
Degree 22
Conductor 1313
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.7·2-s + 250.·3-s − 123.·4-s + 1.55e3·5-s + 4.93e3·6-s − 8.32e3·7-s − 1.25e4·8-s + 4.30e4·9-s + 3.06e4·10-s + 3.07e4·11-s − 3.08e4·12-s + 2.85e4·13-s − 1.64e5·14-s + 3.89e5·15-s − 1.83e5·16-s − 6.37e5·17-s + 8.48e5·18-s + 1.05e5·19-s − 1.91e5·20-s − 2.08e6·21-s + 6.06e5·22-s − 5.11e5·23-s − 3.13e6·24-s + 4.66e5·25-s + 5.63e5·26-s + 5.85e6·27-s + 1.02e6·28-s + ⋯
L(s)  = 1  + 0.871·2-s + 1.78·3-s − 0.240·4-s + 1.11·5-s + 1.55·6-s − 1.31·7-s − 1.08·8-s + 2.18·9-s + 0.969·10-s + 0.633·11-s − 0.429·12-s + 0.277·13-s − 1.14·14-s + 1.98·15-s − 0.701·16-s − 1.85·17-s + 1.90·18-s + 0.186·19-s − 0.267·20-s − 2.34·21-s + 0.551·22-s − 0.380·23-s − 1.93·24-s + 0.238·25-s + 0.241·26-s + 2.12·27-s + 0.315·28-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: 11
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: 00
Selberg data: (2, 13, ( :9/2), 1)(2,\ 13,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 3.7035237283.703523728
L(12)L(\frac12) ≈\approx 3.7035237283.703523728
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 1−19.7T+512T2 1 - 19.7T + 512T^{2}
3 1−250.T+1.96e4T2 1 - 250.T + 1.96e4T^{2}
5 1−1.55e3T+1.95e6T2 1 - 1.55e3T + 1.95e6T^{2}
7 1+8.32e3T+4.03e7T2 1 + 8.32e3T + 4.03e7T^{2}
11 1−3.07e4T+2.35e9T2 1 - 3.07e4T + 2.35e9T^{2}
17 1+6.37e5T+1.18e11T2 1 + 6.37e5T + 1.18e11T^{2}
19 1−1.05e5T+3.22e11T2 1 - 1.05e5T + 3.22e11T^{2}
23 1+5.11e5T+1.80e12T2 1 + 5.11e5T + 1.80e12T^{2}
29 1−7.81e5T+1.45e13T2 1 - 7.81e5T + 1.45e13T^{2}
31 1+2.83e6T+2.64e13T2 1 + 2.83e6T + 2.64e13T^{2}
37 1−1.22e7T+1.29e14T2 1 - 1.22e7T + 1.29e14T^{2}
41 1+6.83e6T+3.27e14T2 1 + 6.83e6T + 3.27e14T^{2}
43 1−3.84e7T+5.02e14T2 1 - 3.84e7T + 5.02e14T^{2}
47 1+1.30e7T+1.11e15T2 1 + 1.30e7T + 1.11e15T^{2}
53 1+2.42e7T+3.29e15T2 1 + 2.42e7T + 3.29e15T^{2}
59 1−1.63e8T+8.66e15T2 1 - 1.63e8T + 8.66e15T^{2}
61 1−1.90e7T+1.16e16T2 1 - 1.90e7T + 1.16e16T^{2}
67 1+7.22e7T+2.72e16T2 1 + 7.22e7T + 2.72e16T^{2}
71 1−2.65e7T+4.58e16T2 1 - 2.65e7T + 4.58e16T^{2}
73 1−2.42e8T+5.88e16T2 1 - 2.42e8T + 5.88e16T^{2}
79 1+4.64e8T+1.19e17T2 1 + 4.64e8T + 1.19e17T^{2}
83 1−5.46e8T+1.86e17T2 1 - 5.46e8T + 1.86e17T^{2}
89 1−3.65e8T+3.50e17T2 1 - 3.65e8T + 3.50e17T^{2}
97 1−9.98e7T+7.60e17T2 1 - 9.98e7T + 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

−17.94519099041213203106896708155, −15.74541872677088976201740041247, −14.45482007432115288086510408978, −13.49504937934845821201429638115, −12.95942396114845730140325352163, −9.640976171901390060997929183005, −8.955033076950461752009246256058, −6.43043112468585301663273465919, −3.92162465517986363555414252976, −2.48308230655511526929776622450, 2.48308230655511526929776622450, 3.92162465517986363555414252976, 6.43043112468585301663273465919, 8.955033076950461752009246256058, 9.640976171901390060997929183005, 12.95942396114845730140325352163, 13.49504937934845821201429638115, 14.45482007432115288086510408978, 15.74541872677088976201740041247, 17.94519099041213203106896708155

Graph of the ZZ-function along the critical line