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Properties

Label 2-624-1.1-c5-0-18
Degree 22
Conductor 624624
Sign 11
Analytic cond. 100.079100.079
Root an. cond. 10.003910.0039
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  + 9·3-s − 11.6·5-s − 90.1·7-s + 81·9-s + 466.·11-s + 169·13-s − 104.·15-s + 1.39e3·17-s − 1.86e3·19-s − 811.·21-s + 4.16e3·23-s − 2.98e3·25-s + 729·27-s − 3.55e3·29-s + 1.95e3·31-s + 4.19e3·33-s + 1.04e3·35-s − 1.42e4·37-s + 1.52e3·39-s − 3.60e3·41-s − 1.70e3·43-s − 941.·45-s + 1.24e4·47-s − 8.67e3·49-s + 1.25e4·51-s + 2.41e3·53-s − 5.41e3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.207·5-s − 0.695·7-s + 0.333·9-s + 1.16·11-s + 0.277·13-s − 0.120·15-s + 1.16·17-s − 1.18·19-s − 0.401·21-s + 1.64·23-s − 0.956·25-s + 0.192·27-s − 0.785·29-s + 0.364·31-s + 0.670·33-s + 0.144·35-s − 1.71·37-s + 0.160·39-s − 0.334·41-s − 0.140·43-s − 0.0693·45-s + 0.821·47-s − 0.516·49-s + 0.673·51-s + 0.117·53-s − 0.241·55-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 100.079100.079
Root analytic conductor: 10.003910.0039
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 624, ( :5/2), 1)(2,\ 624,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.6372797322.637279732
L(12)L(\frac12) ≈\approx 2.6372797322.637279732
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 1
3 1−9T 1 - 9T
13 1−169T 1 - 169T
good5 1+11.6T+3.12e3T2 1 + 11.6T + 3.12e3T^{2}
7 1+90.1T+1.68e4T2 1 + 90.1T + 1.68e4T^{2}
11 1−466.T+1.61e5T2 1 - 466.T + 1.61e5T^{2}
17 1−1.39e3T+1.41e6T2 1 - 1.39e3T + 1.41e6T^{2}
19 1+1.86e3T+2.47e6T2 1 + 1.86e3T + 2.47e6T^{2}
23 1−4.16e3T+6.43e6T2 1 - 4.16e3T + 6.43e6T^{2}
29 1+3.55e3T+2.05e7T2 1 + 3.55e3T + 2.05e7T^{2}
31 1−1.95e3T+2.86e7T2 1 - 1.95e3T + 2.86e7T^{2}
37 1+1.42e4T+6.93e7T2 1 + 1.42e4T + 6.93e7T^{2}
41 1+3.60e3T+1.15e8T2 1 + 3.60e3T + 1.15e8T^{2}
43 1+1.70e3T+1.47e8T2 1 + 1.70e3T + 1.47e8T^{2}
47 1−1.24e4T+2.29e8T2 1 - 1.24e4T + 2.29e8T^{2}
53 1−2.41e3T+4.18e8T2 1 - 2.41e3T + 4.18e8T^{2}
59 1−4.90e4T+7.14e8T2 1 - 4.90e4T + 7.14e8T^{2}
61 1−1.73e4T+8.44e8T2 1 - 1.73e4T + 8.44e8T^{2}
67 1−5.30e4T+1.35e9T2 1 - 5.30e4T + 1.35e9T^{2}
71 1+3.24e4T+1.80e9T2 1 + 3.24e4T + 1.80e9T^{2}
73 1−7.20e4T+2.07e9T2 1 - 7.20e4T + 2.07e9T^{2}
79 1−4.72e4T+3.07e9T2 1 - 4.72e4T + 3.07e9T^{2}
83 1−6.48e4T+3.93e9T2 1 - 6.48e4T + 3.93e9T^{2}
89 1−8.85e4T+5.58e9T2 1 - 8.85e4T + 5.58e9T^{2}
97 1−4.66e4T+8.58e9T2 1 - 4.66e4T + 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

−9.689488232120836115931287947870, −8.992114994323481478423055003128, −8.212887225612001405273547203524, −7.10187771550064738202608344930, −6.44625700501869016340572445823, −5.26248305000633231775992636285, −3.89059007764416014080916947519, −3.37449818381322339712164425738, −1.99537174289807526117313582245, −0.77136800194101176246172206176, 0.77136800194101176246172206176, 1.99537174289807526117313582245, 3.37449818381322339712164425738, 3.89059007764416014080916947519, 5.26248305000633231775992636285, 6.44625700501869016340572445823, 7.10187771550064738202608344930, 8.212887225612001405273547203524, 8.992114994323481478423055003128, 9.689488232120836115931287947870

Graph of the ZZ-function along the critical line