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Properties

Label 2-624-1.1-c5-0-17
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 − 67.5·5-s + 95.5·7-s + 81·9-s + 358.·11-s − 169·13-s − 608.·15-s − 1.90e3·17-s + 1.57e3·19-s + 860.·21-s + 1.51e3·23-s + 1.44e3·25-s + 729·27-s − 2.98e3·29-s + 3.79e3·31-s + 3.22e3·33-s − 6.46e3·35-s + 4.46e3·37-s − 1.52e3·39-s + 5.28e3·41-s − 1.01e4·43-s − 5.47e3·45-s + 6.97e3·47-s − 7.67e3·49-s − 1.71e4·51-s − 9.15e3·53-s − 2.42e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.20·5-s + 0.737·7-s + 0.333·9-s + 0.893·11-s − 0.277·13-s − 0.698·15-s − 1.60·17-s + 1.00·19-s + 0.425·21-s + 0.598·23-s + 0.461·25-s + 0.192·27-s − 0.658·29-s + 0.709·31-s + 0.516·33-s − 0.891·35-s + 0.535·37-s − 0.160·39-s + 0.490·41-s − 0.837·43-s − 0.402·45-s + 0.460·47-s − 0.456·49-s − 0.924·51-s − 0.447·53-s − 1.08·55-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 2.3061185872.306118587
L(12)L(\frac12) ≈\approx 2.3061185872.306118587
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+67.5T+3.12e3T2 1 + 67.5T + 3.12e3T^{2}
7 1−95.5T+1.68e4T2 1 - 95.5T + 1.68e4T^{2}
11 1−358.T+1.61e5T2 1 - 358.T + 1.61e5T^{2}
17 1+1.90e3T+1.41e6T2 1 + 1.90e3T + 1.41e6T^{2}
19 1−1.57e3T+2.47e6T2 1 - 1.57e3T + 2.47e6T^{2}
23 1−1.51e3T+6.43e6T2 1 - 1.51e3T + 6.43e6T^{2}
29 1+2.98e3T+2.05e7T2 1 + 2.98e3T + 2.05e7T^{2}
31 1−3.79e3T+2.86e7T2 1 - 3.79e3T + 2.86e7T^{2}
37 1−4.46e3T+6.93e7T2 1 - 4.46e3T + 6.93e7T^{2}
41 1−5.28e3T+1.15e8T2 1 - 5.28e3T + 1.15e8T^{2}
43 1+1.01e4T+1.47e8T2 1 + 1.01e4T + 1.47e8T^{2}
47 1−6.97e3T+2.29e8T2 1 - 6.97e3T + 2.29e8T^{2}
53 1+9.15e3T+4.18e8T2 1 + 9.15e3T + 4.18e8T^{2}
59 1+2.55e4T+7.14e8T2 1 + 2.55e4T + 7.14e8T^{2}
61 1−2.66e4T+8.44e8T2 1 - 2.66e4T + 8.44e8T^{2}
67 1−528.T+1.35e9T2 1 - 528.T + 1.35e9T^{2}
71 1−2.11e4T+1.80e9T2 1 - 2.11e4T + 1.80e9T^{2}
73 1−3.20e4T+2.07e9T2 1 - 3.20e4T + 2.07e9T^{2}
79 1−6.92e4T+3.07e9T2 1 - 6.92e4T + 3.07e9T^{2}
83 1−6.54e4T+3.93e9T2 1 - 6.54e4T + 3.93e9T^{2}
89 1+2.90e4T+5.58e9T2 1 + 2.90e4T + 5.58e9T^{2}
97 1−1.23e5T+8.58e9T2 1 - 1.23e5T + 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

−9.608492558131173050674859010110, −8.876938081505045101186637111331, −8.057117528303973454014846336664, −7.37752855187079046669474253676, −6.49304330949271432067933704805, −4.92232796835032584511801343552, −4.20563994946450867803776276154, −3.28840277876028853515307518667, −1.99280869206188197649617503455, −0.71781670754388292509919460849, 0.71781670754388292509919460849, 1.99280869206188197649617503455, 3.28840277876028853515307518667, 4.20563994946450867803776276154, 4.92232796835032584511801343552, 6.49304330949271432067933704805, 7.37752855187079046669474253676, 8.057117528303973454014846336664, 8.876938081505045101186637111331, 9.608492558131173050674859010110

Graph of the ZZ-function along the critical line