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Properties

Label 2-693-1.1-c3-0-20
Degree 22
Conductor 693693
Sign 11
Analytic cond. 40.888340.8883
Root an. cond. 6.394396.39439
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.50·2-s + 4.27·4-s − 15.3·5-s − 7·7-s − 13.0·8-s − 53.6·10-s − 11·11-s + 5.90·13-s − 24.5·14-s − 79.9·16-s + 73.2·17-s + 80.3·19-s − 65.4·20-s − 38.5·22-s + 190.·23-s + 109.·25-s + 20.7·26-s − 29.9·28-s + 285.·29-s + 85.3·31-s − 175.·32-s + 256.·34-s + 107.·35-s − 149.·37-s + 281.·38-s + 199.·40-s − 13.9·41-s + ⋯
L(s)  = 1  + 1.23·2-s + 0.534·4-s − 1.36·5-s − 0.377·7-s − 0.576·8-s − 1.69·10-s − 0.301·11-s + 0.126·13-s − 0.468·14-s − 1.24·16-s + 1.04·17-s + 0.970·19-s − 0.731·20-s − 0.373·22-s + 1.72·23-s + 0.874·25-s + 0.156·26-s − 0.201·28-s + 1.82·29-s + 0.494·31-s − 0.970·32-s + 1.29·34-s + 0.517·35-s − 0.662·37-s + 1.20·38-s + 0.789·40-s − 0.0530·41-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 40.888340.8883
Root analytic conductor: 6.394396.39439
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 693, ( :3/2), 1)(2,\ 693,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.4342863522.434286352
L(12)L(\frac12) ≈\approx 2.4342863522.434286352
L(52)L(\frac{5}{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)
bad3 1 1
7 1+7T 1 + 7T
11 1+11T 1 + 11T
good2 1−3.50T+8T2 1 - 3.50T + 8T^{2}
5 1+15.3T+125T2 1 + 15.3T + 125T^{2}
13 1−5.90T+2.19e3T2 1 - 5.90T + 2.19e3T^{2}
17 1−73.2T+4.91e3T2 1 - 73.2T + 4.91e3T^{2}
19 1−80.3T+6.85e3T2 1 - 80.3T + 6.85e3T^{2}
23 1−190.T+1.21e4T2 1 - 190.T + 1.21e4T^{2}
29 1−285.T+2.43e4T2 1 - 285.T + 2.43e4T^{2}
31 1−85.3T+2.97e4T2 1 - 85.3T + 2.97e4T^{2}
37 1+149.T+5.06e4T2 1 + 149.T + 5.06e4T^{2}
41 1+13.9T+6.89e4T2 1 + 13.9T + 6.89e4T^{2}
43 1+148.T+7.95e4T2 1 + 148.T + 7.95e4T^{2}
47 1+94.7T+1.03e5T2 1 + 94.7T + 1.03e5T^{2}
53 1+105.T+1.48e5T2 1 + 105.T + 1.48e5T^{2}
59 1+184.T+2.05e5T2 1 + 184.T + 2.05e5T^{2}
61 1+189.T+2.26e5T2 1 + 189.T + 2.26e5T^{2}
67 1+193.T+3.00e5T2 1 + 193.T + 3.00e5T^{2}
71 1+356.T+3.57e5T2 1 + 356.T + 3.57e5T^{2}
73 1−655.T+3.89e5T2 1 - 655.T + 3.89e5T^{2}
79 1−110.T+4.93e5T2 1 - 110.T + 4.93e5T^{2}
83 1−1.27e3T+5.71e5T2 1 - 1.27e3T + 5.71e5T^{2}
89 1+255.T+7.04e5T2 1 + 255.T + 7.04e5T^{2}
97 1−372.T+9.12e5T2 1 - 372.T + 9.12e5T^{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

−10.24022825879526835936040370793, −9.121663593155540472888652924189, −8.191935624637423795010025924167, −7.29288316017452578450057342191, −6.41105163973568419056448121556, −5.21988589815245406818198945664, −4.59160597174282663975689284017, −3.40861525127283422849689813249, −3.03748069737443641897987716967, −0.74723587190876114952230456649, 0.74723587190876114952230456649, 3.03748069737443641897987716967, 3.40861525127283422849689813249, 4.59160597174282663975689284017, 5.21988589815245406818198945664, 6.41105163973568419056448121556, 7.29288316017452578450057342191, 8.191935624637423795010025924167, 9.121663593155540472888652924189, 10.24022825879526835936040370793

Graph of the ZZ-function along the critical line