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Properties

Label 2-693-1.1-c1-0-3
Degree 22
Conductor 693693
Sign 11
Analytic cond. 5.533635.53363
Root an. cond. 2.352362.35236
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.52·2-s + 4.39·4-s − 0.133·5-s − 7-s − 6.05·8-s + 0.337·10-s − 11-s + 0.133·13-s + 2.52·14-s + 6.52·16-s + 5.05·17-s − 0.924·19-s − 0.586·20-s + 2.52·22-s + 7.05·23-s − 4.98·25-s − 0.337·26-s − 4.39·28-s − 3.86·29-s + 2.79·31-s − 4.39·32-s − 12.7·34-s + 0.133·35-s + 9.98·37-s + 2.33·38-s + 0.808·40-s − 11.8·41-s + ⋯
L(s)  = 1  − 1.78·2-s + 2.19·4-s − 0.0596·5-s − 0.377·7-s − 2.14·8-s + 0.106·10-s − 0.301·11-s + 0.0370·13-s + 0.675·14-s + 1.63·16-s + 1.22·17-s − 0.212·19-s − 0.131·20-s + 0.539·22-s + 1.47·23-s − 0.996·25-s − 0.0662·26-s − 0.830·28-s − 0.717·29-s + 0.501·31-s − 0.777·32-s − 2.19·34-s + 0.0225·35-s + 1.64·37-s + 0.379·38-s + 0.127·40-s − 1.85·41-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 5.533635.53363
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 693, ( :1/2), 1)(2,\ 693,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.59319173390.5931917339
L(12)L(\frac12) ≈\approx 0.59319173390.5931917339
L(32)L(\frac{3}{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+T 1 + T
11 1+T 1 + T
good2 1+2.52T+2T2 1 + 2.52T + 2T^{2}
5 1+0.133T+5T2 1 + 0.133T + 5T^{2}
13 1−0.133T+13T2 1 - 0.133T + 13T^{2}
17 1−5.05T+17T2 1 - 5.05T + 17T^{2}
19 1+0.924T+19T2 1 + 0.924T + 19T^{2}
23 1−7.05T+23T2 1 - 7.05T + 23T^{2}
29 1+3.86T+29T2 1 + 3.86T + 29T^{2}
31 1−2.79T+31T2 1 - 2.79T + 31T^{2}
37 1−9.98T+37T2 1 - 9.98T + 37T^{2}
41 1+11.8T+41T2 1 + 11.8T + 41T^{2}
43 1+3.05T+43T2 1 + 3.05T + 43T^{2}
47 1−3.07T+47T2 1 - 3.07T + 47T^{2}
53 1−4.79T+53T2 1 - 4.79T + 53T^{2}
59 1−12.6T+59T2 1 - 12.6T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1−8.92T+67T2 1 - 8.92T + 67T^{2}
71 1−6.11T+71T2 1 - 6.11T + 71T^{2}
73 1−7.86T+73T2 1 - 7.86T + 73T^{2}
79 1−14.1T+79T2 1 - 14.1T + 79T^{2}
83 1−1.20T+83T2 1 - 1.20T + 83T^{2}
89 1+15.5T+89T2 1 + 15.5T + 89T^{2}
97 1−12.7T+97T2 1 - 12.7T + 97T^{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

−10.06287083629663070256959805184, −9.742622373161863821386341305810, −8.751931211758680633962335380238, −8.016988979946331964722263858808, −7.26462857320688202185871362338, −6.42103710521066609766729256108, −5.30200701719432773777941487258, −3.50835113085756827866004396710, −2.29477518003028835573674020964, −0.849121879007399067292849754039, 0.849121879007399067292849754039, 2.29477518003028835573674020964, 3.50835113085756827866004396710, 5.30200701719432773777941487258, 6.42103710521066609766729256108, 7.26462857320688202185871362338, 8.016988979946331964722263858808, 8.751931211758680633962335380238, 9.742622373161863821386341305810, 10.06287083629663070256959805184

Graph of the ZZ-function along the critical line