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Properties

Label 2-624-1.1-c5-0-48
Degree 22
Conductor 624624
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s + 94.1·5-s − 33.4·7-s + 81·9-s − 109.·11-s − 169·13-s − 847.·15-s + 1.15e3·17-s − 1.31e3·19-s + 301.·21-s − 52.2·23-s + 5.73e3·25-s − 729·27-s − 7.09e3·29-s − 8.21e3·31-s + 988.·33-s − 3.14e3·35-s − 1.28e4·37-s + 1.52e3·39-s + 2.81e3·41-s − 3.49e3·43-s + 7.62e3·45-s + 4.23e3·47-s − 1.56e4·49-s − 1.03e4·51-s + 1.35e4·53-s − 1.03e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.68·5-s − 0.258·7-s + 0.333·9-s − 0.273·11-s − 0.277·13-s − 0.972·15-s + 0.969·17-s − 0.834·19-s + 0.149·21-s − 0.0205·23-s + 1.83·25-s − 0.192·27-s − 1.56·29-s − 1.53·31-s + 0.157·33-s − 0.434·35-s − 1.53·37-s + 0.160·39-s + 0.261·41-s − 0.288·43-s + 0.561·45-s + 0.279·47-s − 0.933·49-s − 0.559·51-s + 0.661·53-s − 0.460·55-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: −1-1
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: 11
Selberg data: (2, 624, ( :5/2), −1)(2,\ 624,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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 1−94.1T+3.12e3T2 1 - 94.1T + 3.12e3T^{2}
7 1+33.4T+1.68e4T2 1 + 33.4T + 1.68e4T^{2}
11 1+109.T+1.61e5T2 1 + 109.T + 1.61e5T^{2}
17 1−1.15e3T+1.41e6T2 1 - 1.15e3T + 1.41e6T^{2}
19 1+1.31e3T+2.47e6T2 1 + 1.31e3T + 2.47e6T^{2}
23 1+52.2T+6.43e6T2 1 + 52.2T + 6.43e6T^{2}
29 1+7.09e3T+2.05e7T2 1 + 7.09e3T + 2.05e7T^{2}
31 1+8.21e3T+2.86e7T2 1 + 8.21e3T + 2.86e7T^{2}
37 1+1.28e4T+6.93e7T2 1 + 1.28e4T + 6.93e7T^{2}
41 1−2.81e3T+1.15e8T2 1 - 2.81e3T + 1.15e8T^{2}
43 1+3.49e3T+1.47e8T2 1 + 3.49e3T + 1.47e8T^{2}
47 1−4.23e3T+2.29e8T2 1 - 4.23e3T + 2.29e8T^{2}
53 1−1.35e4T+4.18e8T2 1 - 1.35e4T + 4.18e8T^{2}
59 1−4.26e4T+7.14e8T2 1 - 4.26e4T + 7.14e8T^{2}
61 1−3.59e4T+8.44e8T2 1 - 3.59e4T + 8.44e8T^{2}
67 1+3.51e4T+1.35e9T2 1 + 3.51e4T + 1.35e9T^{2}
71 1+8.00e3T+1.80e9T2 1 + 8.00e3T + 1.80e9T^{2}
73 1+1.16e4T+2.07e9T2 1 + 1.16e4T + 2.07e9T^{2}
79 1+2.82e4T+3.07e9T2 1 + 2.82e4T + 3.07e9T^{2}
83 1+3.12e3T+3.93e9T2 1 + 3.12e3T + 3.93e9T^{2}
89 1−9.25e4T+5.58e9T2 1 - 9.25e4T + 5.58e9T^{2}
97 1−6.86e3T+8.58e9T2 1 - 6.86e3T + 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

−9.578908631620483307790933046459, −8.772635105826026595770429892442, −7.41207973736681211725766963521, −6.54808797305051215125570994080, −5.60378161056747434002530798638, −5.24320583567421173657447011017, −3.69738870379290766701756430400, −2.31381533049260790471630204992, −1.47443454240362464698813516608, 0, 1.47443454240362464698813516608, 2.31381533049260790471630204992, 3.69738870379290766701756430400, 5.24320583567421173657447011017, 5.60378161056747434002530798638, 6.54808797305051215125570994080, 7.41207973736681211725766963521, 8.772635105826026595770429892442, 9.578908631620483307790933046459

Graph of the ZZ-function along the critical line