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Properties

Label 2-624-1.1-c5-0-44
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 + 26.3·5-s + 108.·7-s + 81·9-s − 285.·11-s + 169·13-s − 237.·15-s + 964.·17-s − 1.12e3·19-s − 977.·21-s − 3.19e3·23-s − 2.43e3·25-s − 729·27-s − 150.·29-s + 1.01e3·31-s + 2.57e3·33-s + 2.86e3·35-s + 1.13e4·37-s − 1.52e3·39-s + 3.40e3·41-s − 1.28e4·43-s + 2.13e3·45-s − 1.95e4·47-s − 5.01e3·49-s − 8.67e3·51-s − 9.70e3·53-s − 7.52e3·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.471·5-s + 0.837·7-s + 0.333·9-s − 0.711·11-s + 0.277·13-s − 0.272·15-s + 0.809·17-s − 0.713·19-s − 0.483·21-s − 1.26·23-s − 0.777·25-s − 0.192·27-s − 0.0332·29-s + 0.190·31-s + 0.410·33-s + 0.394·35-s + 1.35·37-s − 0.160·39-s + 0.315·41-s − 1.05·43-s + 0.157·45-s − 1.29·47-s − 0.298·49-s − 0.467·51-s − 0.474·53-s − 0.335·55-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 1−26.3T+3.12e3T2 1 - 26.3T + 3.12e3T^{2}
7 1−108.T+1.68e4T2 1 - 108.T + 1.68e4T^{2}
11 1+285.T+1.61e5T2 1 + 285.T + 1.61e5T^{2}
17 1−964.T+1.41e6T2 1 - 964.T + 1.41e6T^{2}
19 1+1.12e3T+2.47e6T2 1 + 1.12e3T + 2.47e6T^{2}
23 1+3.19e3T+6.43e6T2 1 + 3.19e3T + 6.43e6T^{2}
29 1+150.T+2.05e7T2 1 + 150.T + 2.05e7T^{2}
31 1−1.01e3T+2.86e7T2 1 - 1.01e3T + 2.86e7T^{2}
37 1−1.13e4T+6.93e7T2 1 - 1.13e4T + 6.93e7T^{2}
41 1−3.40e3T+1.15e8T2 1 - 3.40e3T + 1.15e8T^{2}
43 1+1.28e4T+1.47e8T2 1 + 1.28e4T + 1.47e8T^{2}
47 1+1.95e4T+2.29e8T2 1 + 1.95e4T + 2.29e8T^{2}
53 1+9.70e3T+4.18e8T2 1 + 9.70e3T + 4.18e8T^{2}
59 1−2.27e4T+7.14e8T2 1 - 2.27e4T + 7.14e8T^{2}
61 1−4.14e4T+8.44e8T2 1 - 4.14e4T + 8.44e8T^{2}
67 1−1.53e4T+1.35e9T2 1 - 1.53e4T + 1.35e9T^{2}
71 1+2.98e4T+1.80e9T2 1 + 2.98e4T + 1.80e9T^{2}
73 1−3.76e4T+2.07e9T2 1 - 3.76e4T + 2.07e9T^{2}
79 1+2.80e4T+3.07e9T2 1 + 2.80e4T + 3.07e9T^{2}
83 1−3.05e4T+3.93e9T2 1 - 3.05e4T + 3.93e9T^{2}
89 1−6.65e3T+5.58e9T2 1 - 6.65e3T + 5.58e9T^{2}
97 1+6.86e4T+8.58e9T2 1 + 6.86e4T + 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

−9.685526584328089355798468456868, −8.274042276430446052894306978062, −7.82514455563205165341099015109, −6.52170284928462768188382014341, −5.71776360084898646915058369712, −4.93233942190561517574760165646, −3.87039399529978680144477919958, −2.35252437946620142354283522279, −1.36593773920144369393149596911, 0, 1.36593773920144369393149596911, 2.35252437946620142354283522279, 3.87039399529978680144477919958, 4.93233942190561517574760165646, 5.71776360084898646915058369712, 6.52170284928462768188382014341, 7.82514455563205165341099015109, 8.274042276430446052894306978062, 9.685526584328089355798468456868

Graph of the ZZ-function along the critical line