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Properties

Label 2-13-1.1-c5-0-0
Degree 22
Conductor 1313
Sign 11
Analytic cond. 2.084982.08498
Root an. cond. 1.443941.44394
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.64·2-s + 10.2·3-s + 42.7·4-s + 92.1·5-s − 88.9·6-s + 2.86·7-s − 93.3·8-s − 137.·9-s − 797.·10-s + 571.·11-s + 440.·12-s + 169·13-s − 24.7·14-s + 948.·15-s − 561.·16-s − 855.·17-s + 1.18e3·18-s − 2.35e3·19-s + 3.94e3·20-s + 29.4·21-s − 4.94e3·22-s + 2.50e3·23-s − 960.·24-s + 5.37e3·25-s − 1.46e3·26-s − 3.91e3·27-s + 122.·28-s + ⋯
L(s)  = 1  − 1.52·2-s + 0.659·3-s + 1.33·4-s + 1.64·5-s − 1.00·6-s + 0.0220·7-s − 0.515·8-s − 0.564·9-s − 2.52·10-s + 1.42·11-s + 0.882·12-s + 0.277·13-s − 0.0337·14-s + 1.08·15-s − 0.548·16-s − 0.718·17-s + 0.863·18-s − 1.49·19-s + 2.20·20-s + 0.0145·21-s − 2.17·22-s + 0.989·23-s − 0.340·24-s + 1.71·25-s − 0.424·26-s − 1.03·27-s + 0.0295·28-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 11
Analytic conductor: 2.084982.08498
Root analytic conductor: 1.443941.44394
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 13, ( :5/2), 1)(2,\ 13,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.92402666620.9240266662
L(12)L(\frac12) ≈\approx 0.92402666620.9240266662
L(72)L(\frac{7}{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 1−169T 1 - 169T
good2 1+8.64T+32T2 1 + 8.64T + 32T^{2}
3 1−10.2T+243T2 1 - 10.2T + 243T^{2}
5 1−92.1T+3.12e3T2 1 - 92.1T + 3.12e3T^{2}
7 1−2.86T+1.68e4T2 1 - 2.86T + 1.68e4T^{2}
11 1−571.T+1.61e5T2 1 - 571.T + 1.61e5T^{2}
17 1+855.T+1.41e6T2 1 + 855.T + 1.41e6T^{2}
19 1+2.35e3T+2.47e6T2 1 + 2.35e3T + 2.47e6T^{2}
23 1−2.50e3T+6.43e6T2 1 - 2.50e3T + 6.43e6T^{2}
29 1+5.49e3T+2.05e7T2 1 + 5.49e3T + 2.05e7T^{2}
31 1+144.T+2.86e7T2 1 + 144.T + 2.86e7T^{2}
37 1+515.T+6.93e7T2 1 + 515.T + 6.93e7T^{2}
41 1+1.39e4T+1.15e8T2 1 + 1.39e4T + 1.15e8T^{2}
43 1−7.75e3T+1.47e8T2 1 - 7.75e3T + 1.47e8T^{2}
47 1−8.34e3T+2.29e8T2 1 - 8.34e3T + 2.29e8T^{2}
53 1+5.97e3T+4.18e8T2 1 + 5.97e3T + 4.18e8T^{2}
59 1−2.11e3T+7.14e8T2 1 - 2.11e3T + 7.14e8T^{2}
61 1+1.73e4T+8.44e8T2 1 + 1.73e4T + 8.44e8T^{2}
67 1+3.12e3T+1.35e9T2 1 + 3.12e3T + 1.35e9T^{2}
71 1−4.33e4T+1.80e9T2 1 - 4.33e4T + 1.80e9T^{2}
73 1−6.80e3T+2.07e9T2 1 - 6.80e3T + 2.07e9T^{2}
79 1+1.28e3T+3.07e9T2 1 + 1.28e3T + 3.07e9T^{2}
83 1−7.14e3T+3.93e9T2 1 - 7.14e3T + 3.93e9T^{2}
89 1−1.07e5T+5.58e9T2 1 - 1.07e5T + 5.58e9T^{2}
97 1−4.24e4T+8.58e9T2 1 - 4.24e4T + 8.58e9T^{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

−18.78632808004585341088647532785, −17.33881633245751868208960812130, −16.99881254617018815196141544414, −14.72974340089338398002793793521, −13.43842315411327882951041609900, −10.94201047970936418269581319864, −9.402071326705819612755137670060, −8.753036729332002384396808079993, −6.49233416537680245637163048603, −1.90911682477042387799092390305, 1.90911682477042387799092390305, 6.49233416537680245637163048603, 8.753036729332002384396808079993, 9.402071326705819612755137670060, 10.94201047970936418269581319864, 13.43842315411327882951041609900, 14.72974340089338398002793793521, 16.99881254617018815196141544414, 17.33881633245751868208960812130, 18.78632808004585341088647532785

Graph of the ZZ-function along the critical line