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Properties

Label 2-588-1.1-c5-0-4
Degree 22
Conductor 588588
Sign 11
Analytic cond. 94.305694.3056
Root an. cond. 9.711119.71111
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 + 46.1·5-s + 81·9-s − 631.·11-s − 1.07e3·13-s − 415.·15-s + 161.·17-s + 1.17e3·19-s + 2.16e3·23-s − 998.·25-s − 729·27-s − 4.49e3·29-s + 318.·31-s + 5.68e3·33-s + 1.51e4·37-s + 9.71e3·39-s + 2.05e4·41-s − 455.·43-s + 3.73e3·45-s − 2.07e4·47-s − 1.45e3·51-s − 1.93e4·53-s − 2.91e4·55-s − 1.05e4·57-s + 6.36e3·59-s − 4.91e4·61-s − 4.97e4·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.824·5-s + 0.333·9-s − 1.57·11-s − 1.77·13-s − 0.476·15-s + 0.135·17-s + 0.747·19-s + 0.852·23-s − 0.319·25-s − 0.192·27-s − 0.991·29-s + 0.0594·31-s + 0.908·33-s + 1.82·37-s + 1.02·39-s + 1.91·41-s − 0.0375·43-s + 0.274·45-s − 1.37·47-s − 0.0780·51-s − 0.943·53-s − 1.29·55-s − 0.431·57-s + 0.238·59-s − 1.69·61-s − 1.46·65-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 11
Analytic conductor: 94.305694.3056
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 588, ( :5/2), 1)(2,\ 588,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.3569802201.356980220
L(12)L(\frac12) ≈\approx 1.3569802201.356980220
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
7 1 1
good5 1−46.1T+3.12e3T2 1 - 46.1T + 3.12e3T^{2}
11 1+631.T+1.61e5T2 1 + 631.T + 1.61e5T^{2}
13 1+1.07e3T+3.71e5T2 1 + 1.07e3T + 3.71e5T^{2}
17 1−161.T+1.41e6T2 1 - 161.T + 1.41e6T^{2}
19 1−1.17e3T+2.47e6T2 1 - 1.17e3T + 2.47e6T^{2}
23 1−2.16e3T+6.43e6T2 1 - 2.16e3T + 6.43e6T^{2}
29 1+4.49e3T+2.05e7T2 1 + 4.49e3T + 2.05e7T^{2}
31 1−318.T+2.86e7T2 1 - 318.T + 2.86e7T^{2}
37 1−1.51e4T+6.93e7T2 1 - 1.51e4T + 6.93e7T^{2}
41 1−2.05e4T+1.15e8T2 1 - 2.05e4T + 1.15e8T^{2}
43 1+455.T+1.47e8T2 1 + 455.T + 1.47e8T^{2}
47 1+2.07e4T+2.29e8T2 1 + 2.07e4T + 2.29e8T^{2}
53 1+1.93e4T+4.18e8T2 1 + 1.93e4T + 4.18e8T^{2}
59 1−6.36e3T+7.14e8T2 1 - 6.36e3T + 7.14e8T^{2}
61 1+4.91e4T+8.44e8T2 1 + 4.91e4T + 8.44e8T^{2}
67 1−3.40e4T+1.35e9T2 1 - 3.40e4T + 1.35e9T^{2}
71 1−6.29e4T+1.80e9T2 1 - 6.29e4T + 1.80e9T^{2}
73 1+8.86e3T+2.07e9T2 1 + 8.86e3T + 2.07e9T^{2}
79 1−3.44e4T+3.07e9T2 1 - 3.44e4T + 3.07e9T^{2}
83 1+7.04e3T+3.93e9T2 1 + 7.04e3T + 3.93e9T^{2}
89 1+2.02e4T+5.58e9T2 1 + 2.02e4T + 5.58e9T^{2}
97 1−5.40e4T+8.58e9T2 1 - 5.40e4T + 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

−9.724223006547540992327786473008, −9.540564798091820657595075232742, −7.84431425494172624980308669078, −7.37270635686067023315511277723, −6.09756040032114426162042649551, −5.30618578983063020268865576510, −4.68324149007996098994271096569, −2.93010157898980910543318897937, −2.07640048635663732507483068213, −0.56103016638208437468744430182, 0.56103016638208437468744430182, 2.07640048635663732507483068213, 2.93010157898980910543318897937, 4.68324149007996098994271096569, 5.30618578983063020268865576510, 6.09756040032114426162042649551, 7.37270635686067023315511277723, 7.84431425494172624980308669078, 9.540564798091820657595075232742, 9.724223006547540992327786473008

Graph of the ZZ-function along the critical line