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Properties

Label 2-693-1.1-c3-0-29
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  + 3.09·2-s + 1.56·4-s − 0.265·5-s + 7·7-s − 19.9·8-s − 0.820·10-s + 11·11-s + 9.99·13-s + 21.6·14-s − 74.0·16-s + 103.·17-s + 54.7·19-s − 0.415·20-s + 34.0·22-s − 26.6·23-s − 124.·25-s + 30.9·26-s + 10.9·28-s − 17.2·29-s + 202.·31-s − 69.8·32-s + 320.·34-s − 1.85·35-s + 244.·37-s + 169.·38-s + 5.28·40-s + 306.·41-s + ⋯
L(s)  = 1  + 1.09·2-s + 0.195·4-s − 0.0237·5-s + 0.377·7-s − 0.879·8-s − 0.0259·10-s + 0.301·11-s + 0.213·13-s + 0.413·14-s − 1.15·16-s + 1.47·17-s + 0.660·19-s − 0.00464·20-s + 0.329·22-s − 0.241·23-s − 0.999·25-s + 0.233·26-s + 0.0739·28-s − 0.110·29-s + 1.17·31-s − 0.385·32-s + 1.61·34-s − 0.00897·35-s + 1.08·37-s + 0.722·38-s + 0.0208·40-s + 1.16·41-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 3.4953160223.495316022
L(12)L(\frac12) ≈\approx 3.4953160223.495316022
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 1−3.09T+8T2 1 - 3.09T + 8T^{2}
5 1+0.265T+125T2 1 + 0.265T + 125T^{2}
13 1−9.99T+2.19e3T2 1 - 9.99T + 2.19e3T^{2}
17 1−103.T+4.91e3T2 1 - 103.T + 4.91e3T^{2}
19 1−54.7T+6.85e3T2 1 - 54.7T + 6.85e3T^{2}
23 1+26.6T+1.21e4T2 1 + 26.6T + 1.21e4T^{2}
29 1+17.2T+2.43e4T2 1 + 17.2T + 2.43e4T^{2}
31 1−202.T+2.97e4T2 1 - 202.T + 2.97e4T^{2}
37 1−244.T+5.06e4T2 1 - 244.T + 5.06e4T^{2}
41 1−306.T+6.89e4T2 1 - 306.T + 6.89e4T^{2}
43 1−330.T+7.95e4T2 1 - 330.T + 7.95e4T^{2}
47 1−74.6T+1.03e5T2 1 - 74.6T + 1.03e5T^{2}
53 1−428.T+1.48e5T2 1 - 428.T + 1.48e5T^{2}
59 1+350.T+2.05e5T2 1 + 350.T + 2.05e5T^{2}
61 1−153.T+2.26e5T2 1 - 153.T + 2.26e5T^{2}
67 1−192.T+3.00e5T2 1 - 192.T + 3.00e5T^{2}
71 1−821.T+3.57e5T2 1 - 821.T + 3.57e5T^{2}
73 1+727.T+3.89e5T2 1 + 727.T + 3.89e5T^{2}
79 1+410.T+4.93e5T2 1 + 410.T + 4.93e5T^{2}
83 1+289.T+5.71e5T2 1 + 289.T + 5.71e5T^{2}
89 1+225.T+7.04e5T2 1 + 225.T + 7.04e5T^{2}
97 1+420.T+9.12e5T2 1 + 420.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

−10.00839544703775419088322859019, −9.314717284416127258382817733924, −8.217696754652972852149057034554, −7.39764088670882584389900141373, −6.07965505672977283676181089883, −5.56302377624991190886534377699, −4.47366350300789552851419321658, −3.69109725887804620821313044823, −2.61092905111339169472971664367, −0.960717694687656106476898507657, 0.960717694687656106476898507657, 2.61092905111339169472971664367, 3.69109725887804620821313044823, 4.47366350300789552851419321658, 5.56302377624991190886534377699, 6.07965505672977283676181089883, 7.39764088670882584389900141373, 8.217696754652972852149057034554, 9.314717284416127258382817733924, 10.00839544703775419088322859019

Graph of the ZZ-function along the critical line