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Properties

Label 2-624-1.1-c5-0-22
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 + 47.5·5-s − 19.5·7-s + 81·9-s − 562.·11-s − 169·13-s + 428.·15-s + 1.08e3·17-s + 1.22e3·19-s − 176.·21-s − 2.16e3·23-s − 860.·25-s + 729·27-s − 2.51e3·29-s + 3.68e3·31-s − 5.06e3·33-s − 931.·35-s + 1.57e4·37-s − 1.52e3·39-s + 1.59e4·41-s + 8.95e3·43-s + 3.85e3·45-s + 3.02e4·47-s − 1.64e4·49-s + 9.78e3·51-s + 2.21e4·53-s − 2.67e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.851·5-s − 0.151·7-s + 0.333·9-s − 1.40·11-s − 0.277·13-s + 0.491·15-s + 0.912·17-s + 0.781·19-s − 0.0872·21-s − 0.854·23-s − 0.275·25-s + 0.192·27-s − 0.556·29-s + 0.687·31-s − 0.809·33-s − 0.128·35-s + 1.89·37-s − 0.160·39-s + 1.48·41-s + 0.738·43-s + 0.283·45-s + 1.99·47-s − 0.977·49-s + 0.526·51-s + 1.08·53-s − 1.19·55-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 3.0623307223.062330722
L(12)L(\frac12) ≈\approx 3.0623307223.062330722
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 1−47.5T+3.12e3T2 1 - 47.5T + 3.12e3T^{2}
7 1+19.5T+1.68e4T2 1 + 19.5T + 1.68e4T^{2}
11 1+562.T+1.61e5T2 1 + 562.T + 1.61e5T^{2}
17 1−1.08e3T+1.41e6T2 1 - 1.08e3T + 1.41e6T^{2}
19 1−1.22e3T+2.47e6T2 1 - 1.22e3T + 2.47e6T^{2}
23 1+2.16e3T+6.43e6T2 1 + 2.16e3T + 6.43e6T^{2}
29 1+2.51e3T+2.05e7T2 1 + 2.51e3T + 2.05e7T^{2}
31 1−3.68e3T+2.86e7T2 1 - 3.68e3T + 2.86e7T^{2}
37 1−1.57e4T+6.93e7T2 1 - 1.57e4T + 6.93e7T^{2}
41 1−1.59e4T+1.15e8T2 1 - 1.59e4T + 1.15e8T^{2}
43 1−8.95e3T+1.47e8T2 1 - 8.95e3T + 1.47e8T^{2}
47 1−3.02e4T+2.29e8T2 1 - 3.02e4T + 2.29e8T^{2}
53 1−2.21e4T+4.18e8T2 1 - 2.21e4T + 4.18e8T^{2}
59 1+2.02e4T+7.14e8T2 1 + 2.02e4T + 7.14e8T^{2}
61 1−1.28e4T+8.44e8T2 1 - 1.28e4T + 8.44e8T^{2}
67 1−3.95e4T+1.35e9T2 1 - 3.95e4T + 1.35e9T^{2}
71 1+6.63e4T+1.80e9T2 1 + 6.63e4T + 1.80e9T^{2}
73 1−7.09e4T+2.07e9T2 1 - 7.09e4T + 2.07e9T^{2}
79 1+9.61e4T+3.07e9T2 1 + 9.61e4T + 3.07e9T^{2}
83 1+4.81e4T+3.93e9T2 1 + 4.81e4T + 3.93e9T^{2}
89 1−8.18e4T+5.58e9T2 1 - 8.18e4T + 5.58e9T^{2}
97 1−2.88e4T+8.58e9T2 1 - 2.88e4T + 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

−9.864252781811406396656993083219, −9.134297168690218187016180482060, −7.87255080306656232349773507671, −7.52327407606657515501047539698, −6.02723164365227113170495039285, −5.44518057073700911663592432471, −4.19949997603866835152391238386, −2.88515947973490616583990901343, −2.21019134427601685390964369008, −0.815595732479606924417724901749, 0.815595732479606924417724901749, 2.21019134427601685390964369008, 2.88515947973490616583990901343, 4.19949997603866835152391238386, 5.44518057073700911663592432471, 6.02723164365227113170495039285, 7.52327407606657515501047539698, 7.87255080306656232349773507671, 9.134297168690218187016180482060, 9.864252781811406396656993083219

Graph of the ZZ-function along the critical line