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Properties

Label 2-693-1.1-c3-0-11
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  − 4.60·2-s + 13.1·4-s − 1.84·5-s − 7·7-s − 23.9·8-s + 8.49·10-s + 11·11-s + 24.6·13-s + 32.2·14-s + 4.57·16-s − 17.8·17-s + 32.1·19-s − 24.3·20-s − 50.6·22-s − 14.1·23-s − 121.·25-s − 113.·26-s − 92.3·28-s + 41.5·29-s + 175.·31-s + 170.·32-s + 82.3·34-s + 12.9·35-s + 292.·37-s − 147.·38-s + 44.1·40-s − 154.·41-s + ⋯
L(s)  = 1  − 1.62·2-s + 1.64·4-s − 0.164·5-s − 0.377·7-s − 1.05·8-s + 0.268·10-s + 0.301·11-s + 0.525·13-s + 0.615·14-s + 0.0714·16-s − 0.255·17-s + 0.388·19-s − 0.272·20-s − 0.490·22-s − 0.128·23-s − 0.972·25-s − 0.855·26-s − 0.623·28-s + 0.266·29-s + 1.01·31-s + 0.940·32-s + 0.415·34-s + 0.0623·35-s + 1.30·37-s − 0.631·38-s + 0.174·40-s − 0.587·41-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 0.70343300460.7034330046
L(12)L(\frac12) ≈\approx 0.70343300460.7034330046
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+4.60T+8T2 1 + 4.60T + 8T^{2}
5 1+1.84T+125T2 1 + 1.84T + 125T^{2}
13 1−24.6T+2.19e3T2 1 - 24.6T + 2.19e3T^{2}
17 1+17.8T+4.91e3T2 1 + 17.8T + 4.91e3T^{2}
19 1−32.1T+6.85e3T2 1 - 32.1T + 6.85e3T^{2}
23 1+14.1T+1.21e4T2 1 + 14.1T + 1.21e4T^{2}
29 1−41.5T+2.43e4T2 1 - 41.5T + 2.43e4T^{2}
31 1−175.T+2.97e4T2 1 - 175.T + 2.97e4T^{2}
37 1−292.T+5.06e4T2 1 - 292.T + 5.06e4T^{2}
41 1+154.T+6.89e4T2 1 + 154.T + 6.89e4T^{2}
43 1+277.T+7.95e4T2 1 + 277.T + 7.95e4T^{2}
47 1−52.1T+1.03e5T2 1 - 52.1T + 1.03e5T^{2}
53 1+82.3T+1.48e5T2 1 + 82.3T + 1.48e5T^{2}
59 1+712.T+2.05e5T2 1 + 712.T + 2.05e5T^{2}
61 1+647.T+2.26e5T2 1 + 647.T + 2.26e5T^{2}
67 1−260.T+3.00e5T2 1 - 260.T + 3.00e5T^{2}
71 1+369.T+3.57e5T2 1 + 369.T + 3.57e5T^{2}
73 1−1.14e3T+3.89e5T2 1 - 1.14e3T + 3.89e5T^{2}
79 1−488.T+4.93e5T2 1 - 488.T + 4.93e5T^{2}
83 1+548.T+5.71e5T2 1 + 548.T + 5.71e5T^{2}
89 1+105.T+7.04e5T2 1 + 105.T + 7.04e5T^{2}
97 1+1.36e3T+9.12e5T2 1 + 1.36e3T + 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

−9.834714463944419357341935098598, −9.300483794889184260812285726285, −8.364852092972904968493954779573, −7.76486902083095688341135759376, −6.76168747437624095766254531347, −6.03114225855345079802739055946, −4.46762749682550905386064060362, −3.12106913975693130489990286922, −1.79913066484315486069267832466, −0.62416324217353957190919860818, 0.62416324217353957190919860818, 1.79913066484315486069267832466, 3.12106913975693130489990286922, 4.46762749682550905386064060362, 6.03114225855345079802739055946, 6.76168747437624095766254531347, 7.76486902083095688341135759376, 8.364852092972904968493954779573, 9.300483794889184260812285726285, 9.834714463944419357341935098598

Graph of the ZZ-function along the critical line