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Properties

Label 2-624-1.1-c5-0-0
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 − 70.9·5-s + 28.4·7-s + 81·9-s − 594.·11-s − 169·13-s + 638.·15-s + 971.·17-s − 1.66e3·19-s − 256.·21-s − 3.44e3·23-s + 1.91e3·25-s − 729·27-s − 964.·29-s − 7.88e3·31-s + 5.35e3·33-s − 2.02e3·35-s − 6.34e3·37-s + 1.52e3·39-s + 2.31e3·41-s − 6.40e3·43-s − 5.74e3·45-s + 1.04e4·47-s − 1.59e4·49-s − 8.74e3·51-s + 8.42e3·53-s + 4.22e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.26·5-s + 0.219·7-s + 0.333·9-s − 1.48·11-s − 0.277·13-s + 0.732·15-s + 0.815·17-s − 1.06·19-s − 0.126·21-s − 1.35·23-s + 0.611·25-s − 0.192·27-s − 0.212·29-s − 1.47·31-s + 0.855·33-s − 0.278·35-s − 0.761·37-s + 0.160·39-s + 0.215·41-s − 0.527·43-s − 0.423·45-s + 0.687·47-s − 0.951·49-s − 0.470·51-s + 0.411·53-s + 1.88·55-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 0.21328401690.2132840169
L(12)L(\frac12) ≈\approx 0.21328401690.2132840169
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+70.9T+3.12e3T2 1 + 70.9T + 3.12e3T^{2}
7 1−28.4T+1.68e4T2 1 - 28.4T + 1.68e4T^{2}
11 1+594.T+1.61e5T2 1 + 594.T + 1.61e5T^{2}
17 1−971.T+1.41e6T2 1 - 971.T + 1.41e6T^{2}
19 1+1.66e3T+2.47e6T2 1 + 1.66e3T + 2.47e6T^{2}
23 1+3.44e3T+6.43e6T2 1 + 3.44e3T + 6.43e6T^{2}
29 1+964.T+2.05e7T2 1 + 964.T + 2.05e7T^{2}
31 1+7.88e3T+2.86e7T2 1 + 7.88e3T + 2.86e7T^{2}
37 1+6.34e3T+6.93e7T2 1 + 6.34e3T + 6.93e7T^{2}
41 1−2.31e3T+1.15e8T2 1 - 2.31e3T + 1.15e8T^{2}
43 1+6.40e3T+1.47e8T2 1 + 6.40e3T + 1.47e8T^{2}
47 1−1.04e4T+2.29e8T2 1 - 1.04e4T + 2.29e8T^{2}
53 1−8.42e3T+4.18e8T2 1 - 8.42e3T + 4.18e8T^{2}
59 1+4.10e4T+7.14e8T2 1 + 4.10e4T + 7.14e8T^{2}
61 1+3.61e4T+8.44e8T2 1 + 3.61e4T + 8.44e8T^{2}
67 1+4.51e4T+1.35e9T2 1 + 4.51e4T + 1.35e9T^{2}
71 1−1.54e4T+1.80e9T2 1 - 1.54e4T + 1.80e9T^{2}
73 1+2.19e4T+2.07e9T2 1 + 2.19e4T + 2.07e9T^{2}
79 1+1.85e4T+3.07e9T2 1 + 1.85e4T + 3.07e9T^{2}
83 1−4.31e4T+3.93e9T2 1 - 4.31e4T + 3.93e9T^{2}
89 1−8.29e4T+5.58e9T2 1 - 8.29e4T + 5.58e9T^{2}
97 1−5.48e4T+8.58e9T2 1 - 5.48e4T + 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

−10.15447239096696604575527800624, −8.813085063696632828508755520452, −7.71899694170262651233710121011, −7.58255626770784725965165423506, −6.15391024494025854576386526788, −5.20353171256444335792694479849, −4.31485966377277382886313562041, −3.31489796375953869853950228936, −1.91113225485209793703329716781, −0.21830398254140576410467392768, 0.21830398254140576410467392768, 1.91113225485209793703329716781, 3.31489796375953869853950228936, 4.31485966377277382886313562041, 5.20353171256444335792694479849, 6.15391024494025854576386526788, 7.58255626770784725965165423506, 7.71899694170262651233710121011, 8.813085063696632828508755520452, 10.15447239096696604575527800624

Graph of the ZZ-function along the critical line