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Properties

Label 2-538-1.1-c7-0-107
Degree 22
Conductor 538538
Sign −1-1
Analytic cond. 168.063168.063
Root an. cond. 12.963912.9639
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 25.5·3-s + 64·4-s + 439.·5-s + 204.·6-s − 363.·7-s − 512·8-s − 1.53e3·9-s − 3.51e3·10-s + 8.20e3·11-s − 1.63e3·12-s − 7.47e3·13-s + 2.91e3·14-s − 1.12e4·15-s + 4.09e3·16-s − 3.15e4·17-s + 1.22e4·18-s + 4.99e4·19-s + 2.81e4·20-s + 9.29e3·21-s − 6.56e4·22-s + 5.76e3·23-s + 1.30e4·24-s + 1.15e5·25-s + 5.98e4·26-s + 9.50e4·27-s − 2.32e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.545·3-s + 0.5·4-s + 1.57·5-s + 0.385·6-s − 0.401·7-s − 0.353·8-s − 0.702·9-s − 1.11·10-s + 1.85·11-s − 0.272·12-s − 0.943·13-s + 0.283·14-s − 0.859·15-s + 0.250·16-s − 1.55·17-s + 0.496·18-s + 1.67·19-s + 0.787·20-s + 0.218·21-s − 1.31·22-s + 0.0988·23-s + 0.192·24-s + 1.47·25-s + 0.667·26-s + 0.929·27-s − 0.200·28-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −1-1
Analytic conductor: 168.063168.063
Root analytic conductor: 12.963912.9639
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 538, ( :7/2), −1)(2,\ 538,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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+8T 1 + 8T
269 1−1.94e7T 1 - 1.94e7T
good3 1+25.5T+2.18e3T2 1 + 25.5T + 2.18e3T^{2}
5 1−439.T+7.81e4T2 1 - 439.T + 7.81e4T^{2}
7 1+363.T+8.23e5T2 1 + 363.T + 8.23e5T^{2}
11 1−8.20e3T+1.94e7T2 1 - 8.20e3T + 1.94e7T^{2}
13 1+7.47e3T+6.27e7T2 1 + 7.47e3T + 6.27e7T^{2}
17 1+3.15e4T+4.10e8T2 1 + 3.15e4T + 4.10e8T^{2}
19 1−4.99e4T+8.93e8T2 1 - 4.99e4T + 8.93e8T^{2}
23 1−5.76e3T+3.40e9T2 1 - 5.76e3T + 3.40e9T^{2}
29 1+1.47e5T+1.72e10T2 1 + 1.47e5T + 1.72e10T^{2}
31 1−1.40e5T+2.75e10T2 1 - 1.40e5T + 2.75e10T^{2}
37 1+4.65e5T+9.49e10T2 1 + 4.65e5T + 9.49e10T^{2}
41 1−6.98e4T+1.94e11T2 1 - 6.98e4T + 1.94e11T^{2}
43 1+2.57e5T+2.71e11T2 1 + 2.57e5T + 2.71e11T^{2}
47 1−8.08e5T+5.06e11T2 1 - 8.08e5T + 5.06e11T^{2}
53 1+5.98e5T+1.17e12T2 1 + 5.98e5T + 1.17e12T^{2}
59 1+1.25e6T+2.48e12T2 1 + 1.25e6T + 2.48e12T^{2}
61 1+1.61e6T+3.14e12T2 1 + 1.61e6T + 3.14e12T^{2}
67 1+2.52e6T+6.06e12T2 1 + 2.52e6T + 6.06e12T^{2}
71 1+1.78e6T+9.09e12T2 1 + 1.78e6T + 9.09e12T^{2}
73 1−5.45e6T+1.10e13T2 1 - 5.45e6T + 1.10e13T^{2}
79 1+4.14e6T+1.92e13T2 1 + 4.14e6T + 1.92e13T^{2}
83 1−1.41e6T+2.71e13T2 1 - 1.41e6T + 2.71e13T^{2}
89 1+1.02e6T+4.42e13T2 1 + 1.02e6T + 4.42e13T^{2}
97 1−7.08e6T+8.07e13T2 1 - 7.08e6T + 8.07e13T^{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

−9.305891543163204427661547000833, −8.826798255620650827029113652472, −7.19246174634155869852945435771, −6.47066773252987709434951656704, −5.85621478342463913805268661568, −4.84172158971979751993903161622, −3.20187107066443377426852700110, −2.09351942524124146508776283799, −1.21493351945459829788566730778, 0, 1.21493351945459829788566730778, 2.09351942524124146508776283799, 3.20187107066443377426852700110, 4.84172158971979751993903161622, 5.85621478342463913805268661568, 6.47066773252987709434951656704, 7.19246174634155869852945435771, 8.826798255620650827029113652472, 9.305891543163204427661547000833

Graph of the ZZ-function along the critical line