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Properties

Label 2-624-1.1-c5-0-1
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 − 62.9·5-s − 85.2·7-s + 81·9-s + 69.5·11-s + 169·13-s + 566.·15-s − 1.37e3·17-s − 1.24e3·19-s + 767.·21-s − 749.·23-s + 843.·25-s − 729·27-s − 2.08e3·29-s − 3.86e3·31-s − 626.·33-s + 5.36e3·35-s − 1.47e4·37-s − 1.52e3·39-s − 1.03e4·41-s − 6.75e3·43-s − 5.10e3·45-s + 6.80e3·47-s − 9.54e3·49-s + 1.23e4·51-s − 2.99e4·53-s − 4.38e3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.12·5-s − 0.657·7-s + 0.333·9-s + 0.173·11-s + 0.277·13-s + 0.650·15-s − 1.15·17-s − 0.788·19-s + 0.379·21-s − 0.295·23-s + 0.269·25-s − 0.192·27-s − 0.459·29-s − 0.722·31-s − 0.100·33-s + 0.740·35-s − 1.76·37-s − 0.160·39-s − 0.965·41-s − 0.557·43-s − 0.375·45-s + 0.449·47-s − 0.567·49-s + 0.665·51-s − 1.46·53-s − 0.195·55-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 0.28298736670.2829873667
L(12)L(\frac12) ≈\approx 0.28298736670.2829873667
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+62.9T+3.12e3T2 1 + 62.9T + 3.12e3T^{2}
7 1+85.2T+1.68e4T2 1 + 85.2T + 1.68e4T^{2}
11 1−69.5T+1.61e5T2 1 - 69.5T + 1.61e5T^{2}
17 1+1.37e3T+1.41e6T2 1 + 1.37e3T + 1.41e6T^{2}
19 1+1.24e3T+2.47e6T2 1 + 1.24e3T + 2.47e6T^{2}
23 1+749.T+6.43e6T2 1 + 749.T + 6.43e6T^{2}
29 1+2.08e3T+2.05e7T2 1 + 2.08e3T + 2.05e7T^{2}
31 1+3.86e3T+2.86e7T2 1 + 3.86e3T + 2.86e7T^{2}
37 1+1.47e4T+6.93e7T2 1 + 1.47e4T + 6.93e7T^{2}
41 1+1.03e4T+1.15e8T2 1 + 1.03e4T + 1.15e8T^{2}
43 1+6.75e3T+1.47e8T2 1 + 6.75e3T + 1.47e8T^{2}
47 1−6.80e3T+2.29e8T2 1 - 6.80e3T + 2.29e8T^{2}
53 1+2.99e4T+4.18e8T2 1 + 2.99e4T + 4.18e8T^{2}
59 1−2.03e3T+7.14e8T2 1 - 2.03e3T + 7.14e8T^{2}
61 1−2.10e4T+8.44e8T2 1 - 2.10e4T + 8.44e8T^{2}
67 1−4.52e4T+1.35e9T2 1 - 4.52e4T + 1.35e9T^{2}
71 1−6.02e4T+1.80e9T2 1 - 6.02e4T + 1.80e9T^{2}
73 1+1.53e3T+2.07e9T2 1 + 1.53e3T + 2.07e9T^{2}
79 1+3.03e4T+3.07e9T2 1 + 3.03e4T + 3.07e9T^{2}
83 1+1.86e4T+3.93e9T2 1 + 1.86e4T + 3.93e9T^{2}
89 1−1.99e4T+5.58e9T2 1 - 1.99e4T + 5.58e9T^{2}
97 1+1.40e5T+8.58e9T2 1 + 1.40e5T + 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

−9.924766961703708938216275619687, −8.890086742764119255999903227311, −8.099336690945009019867822512530, −6.98300354882576335442442866286, −6.44661797476780665264183582937, −5.21910054009892824908019614861, −4.14721621070103243218666131538, −3.43923980181687375625367081222, −1.87505753365457115495476573848, −0.25353645454652869299121311809, 0.25353645454652869299121311809, 1.87505753365457115495476573848, 3.43923980181687375625367081222, 4.14721621070103243218666131538, 5.21910054009892824908019614861, 6.44661797476780665264183582937, 6.98300354882576335442442866286, 8.099336690945009019867822512530, 8.890086742764119255999903227311, 9.924766961703708938216275619687

Graph of the ZZ-function along the critical line