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Properties

Label 2-693-1.1-c3-0-18
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  + 2.18·2-s − 3.20·4-s − 7.60·5-s + 7·7-s − 24.5·8-s − 16.6·10-s − 11·11-s + 0.174·13-s + 15.3·14-s − 28.0·16-s − 128.·17-s + 141.·19-s + 24.4·20-s − 24.0·22-s + 133.·23-s − 67.1·25-s + 0.381·26-s − 22.4·28-s + 177.·29-s + 48.2·31-s + 134.·32-s − 282.·34-s − 53.2·35-s + 161.·37-s + 310.·38-s + 186.·40-s + 195.·41-s + ⋯
L(s)  = 1  + 0.773·2-s − 0.401·4-s − 0.680·5-s + 0.377·7-s − 1.08·8-s − 0.526·10-s − 0.301·11-s + 0.00371·13-s + 0.292·14-s − 0.438·16-s − 1.83·17-s + 1.71·19-s + 0.272·20-s − 0.233·22-s + 1.20·23-s − 0.537·25-s + 0.00287·26-s − 0.151·28-s + 1.13·29-s + 0.279·31-s + 0.745·32-s − 1.42·34-s − 0.257·35-s + 0.718·37-s + 1.32·38-s + 0.737·40-s + 0.745·41-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 1.8957034191.895703419
L(12)L(\frac12) ≈\approx 1.8957034191.895703419
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 1−2.18T+8T2 1 - 2.18T + 8T^{2}
5 1+7.60T+125T2 1 + 7.60T + 125T^{2}
13 1−0.174T+2.19e3T2 1 - 0.174T + 2.19e3T^{2}
17 1+128.T+4.91e3T2 1 + 128.T + 4.91e3T^{2}
19 1−141.T+6.85e3T2 1 - 141.T + 6.85e3T^{2}
23 1−133.T+1.21e4T2 1 - 133.T + 1.21e4T^{2}
29 1−177.T+2.43e4T2 1 - 177.T + 2.43e4T^{2}
31 1−48.2T+2.97e4T2 1 - 48.2T + 2.97e4T^{2}
37 1−161.T+5.06e4T2 1 - 161.T + 5.06e4T^{2}
41 1−195.T+6.89e4T2 1 - 195.T + 6.89e4T^{2}
43 1+488.T+7.95e4T2 1 + 488.T + 7.95e4T^{2}
47 1−171.T+1.03e5T2 1 - 171.T + 1.03e5T^{2}
53 1−431.T+1.48e5T2 1 - 431.T + 1.48e5T^{2}
59 1+194.T+2.05e5T2 1 + 194.T + 2.05e5T^{2}
61 1−585.T+2.26e5T2 1 - 585.T + 2.26e5T^{2}
67 1−155.T+3.00e5T2 1 - 155.T + 3.00e5T^{2}
71 1−374.T+3.57e5T2 1 - 374.T + 3.57e5T^{2}
73 1−210.T+3.89e5T2 1 - 210.T + 3.89e5T^{2}
79 1+7.00T+4.93e5T2 1 + 7.00T + 4.93e5T^{2}
83 1−93.6T+5.71e5T2 1 - 93.6T + 5.71e5T^{2}
89 1−307.T+7.04e5T2 1 - 307.T + 7.04e5T^{2}
97 1−965.T+9.12e5T2 1 - 965.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

−10.05533273953525574010816259034, −9.083977734661292411726894512078, −8.391614100162674214411415807073, −7.39825146645164854647279020770, −6.41434031380007236989548736861, −5.21455232076949395584825089619, −4.61856748588601953014376286832, −3.65767286324615624730066230898, −2.60010869480101745261437697292, −0.70483275033689325434338001405, 0.70483275033689325434338001405, 2.60010869480101745261437697292, 3.65767286324615624730066230898, 4.61856748588601953014376286832, 5.21455232076949395584825089619, 6.41434031380007236989548736861, 7.39825146645164854647279020770, 8.391614100162674214411415807073, 9.083977734661292411726894512078, 10.05533273953525574010816259034

Graph of the ZZ-function along the critical line