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Properties

Label 2-130-1.1-c5-0-8
Degree 22
Conductor 130130
Sign 11
Analytic cond. 20.849820.8498
Root an. cond. 4.566164.56616
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  − 4·2-s + 13.8·3-s + 16·4-s + 25·5-s − 55.2·6-s + 213.·7-s − 64·8-s − 52.2·9-s − 100·10-s + 587.·11-s + 220.·12-s − 169·13-s − 852.·14-s + 345.·15-s + 256·16-s + 162.·17-s + 208.·18-s − 81.3·19-s + 400·20-s + 2.94e3·21-s − 2.34e3·22-s − 2.94e3·23-s − 883.·24-s + 625·25-s + 676·26-s − 4.07e3·27-s + 3.41e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.886·3-s + 0.5·4-s + 0.447·5-s − 0.626·6-s + 1.64·7-s − 0.353·8-s − 0.215·9-s − 0.316·10-s + 1.46·11-s + 0.443·12-s − 0.277·13-s − 1.16·14-s + 0.396·15-s + 0.250·16-s + 0.136·17-s + 0.152·18-s − 0.0517·19-s + 0.223·20-s + 1.45·21-s − 1.03·22-s − 1.16·23-s − 0.313·24-s + 0.200·25-s + 0.196·26-s − 1.07·27-s + 0.822·28-s + ⋯

Functional equation

Λ(s)=(130s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(130s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 130130    =    2⋅5⋅132 \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 20.849820.8498
Root analytic conductor: 4.566164.56616
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 130, ( :5/2), 1)(2,\ 130,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.4624644812.462464481
L(12)L(\frac12) ≈\approx 2.4624644812.462464481
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)
bad2 1+4T 1 + 4T
5 1−25T 1 - 25T
13 1+169T 1 + 169T
good3 1−13.8T+243T2 1 - 13.8T + 243T^{2}
7 1−213.T+1.68e4T2 1 - 213.T + 1.68e4T^{2}
11 1−587.T+1.61e5T2 1 - 587.T + 1.61e5T^{2}
17 1−162.T+1.41e6T2 1 - 162.T + 1.41e6T^{2}
19 1+81.3T+2.47e6T2 1 + 81.3T + 2.47e6T^{2}
23 1+2.94e3T+6.43e6T2 1 + 2.94e3T + 6.43e6T^{2}
29 1−6.25e3T+2.05e7T2 1 - 6.25e3T + 2.05e7T^{2}
31 1+4.03e3T+2.86e7T2 1 + 4.03e3T + 2.86e7T^{2}
37 1−7.61e3T+6.93e7T2 1 - 7.61e3T + 6.93e7T^{2}
41 1−958.T+1.15e8T2 1 - 958.T + 1.15e8T^{2}
43 1−169.T+1.47e8T2 1 - 169.T + 1.47e8T^{2}
47 1−2.16e4T+2.29e8T2 1 - 2.16e4T + 2.29e8T^{2}
53 1−2.47e4T+4.18e8T2 1 - 2.47e4T + 4.18e8T^{2}
59 1−4.01e4T+7.14e8T2 1 - 4.01e4T + 7.14e8T^{2}
61 1+1.63e4T+8.44e8T2 1 + 1.63e4T + 8.44e8T^{2}
67 1−1.83e4T+1.35e9T2 1 - 1.83e4T + 1.35e9T^{2}
71 1+7.78e4T+1.80e9T2 1 + 7.78e4T + 1.80e9T^{2}
73 1+6.21e4T+2.07e9T2 1 + 6.21e4T + 2.07e9T^{2}
79 1+6.05e4T+3.07e9T2 1 + 6.05e4T + 3.07e9T^{2}
83 1+2.65e3T+3.93e9T2 1 + 2.65e3T + 3.93e9T^{2}
89 1−8.22e3T+5.58e9T2 1 - 8.22e3T + 5.58e9T^{2}
97 1+5.38e4T+8.58e9T2 1 + 5.38e4T + 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

−12.05854405541417134351038072232, −11.35304759743360430550470607132, −10.10427462753226056803604770707, −8.968595665529428824116130327796, −8.374126730362824721633275509994, −7.32582695781379449497196645572, −5.82193448033316566986646754901, −4.17040536299388433722684199965, −2.38888123678575678618212379598, −1.30744983143822888102858372711, 1.30744983143822888102858372711, 2.38888123678575678618212379598, 4.17040536299388433722684199965, 5.82193448033316566986646754901, 7.32582695781379449497196645572, 8.374126730362824721633275509994, 8.968595665529428824116130327796, 10.10427462753226056803604770707, 11.35304759743360430550470607132, 12.05854405541417134351038072232

Graph of the ZZ-function along the critical line