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Properties

Label 2-693-1.1-c3-0-4
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  − 3.85·2-s + 6.86·4-s − 5.82·5-s − 7·7-s + 4.38·8-s + 22.4·10-s − 11·11-s − 87.1·13-s + 26.9·14-s − 71.8·16-s + 119.·17-s + 23.9·19-s − 39.9·20-s + 42.4·22-s − 119.·23-s − 91.0·25-s + 335.·26-s − 48.0·28-s + 53.7·29-s − 69.0·31-s + 241.·32-s − 458.·34-s + 40.7·35-s + 28.6·37-s − 92.4·38-s − 25.5·40-s − 419.·41-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.857·4-s − 0.521·5-s − 0.377·7-s + 0.193·8-s + 0.710·10-s − 0.301·11-s − 1.85·13-s + 0.515·14-s − 1.12·16-s + 1.69·17-s + 0.289·19-s − 0.447·20-s + 0.410·22-s − 1.08·23-s − 0.728·25-s + 2.53·26-s − 0.324·28-s + 0.344·29-s − 0.399·31-s + 1.33·32-s − 2.31·34-s + 0.197·35-s + 0.127·37-s − 0.394·38-s − 0.100·40-s − 1.59·41-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 0.39225800250.3922580025
L(12)L(\frac12) ≈\approx 0.39225800250.3922580025
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+3.85T+8T2 1 + 3.85T + 8T^{2}
5 1+5.82T+125T2 1 + 5.82T + 125T^{2}
13 1+87.1T+2.19e3T2 1 + 87.1T + 2.19e3T^{2}
17 1−119.T+4.91e3T2 1 - 119.T + 4.91e3T^{2}
19 1−23.9T+6.85e3T2 1 - 23.9T + 6.85e3T^{2}
23 1+119.T+1.21e4T2 1 + 119.T + 1.21e4T^{2}
29 1−53.7T+2.43e4T2 1 - 53.7T + 2.43e4T^{2}
31 1+69.0T+2.97e4T2 1 + 69.0T + 2.97e4T^{2}
37 1−28.6T+5.06e4T2 1 - 28.6T + 5.06e4T^{2}
41 1+419.T+6.89e4T2 1 + 419.T + 6.89e4T^{2}
43 1+40.0T+7.95e4T2 1 + 40.0T + 7.95e4T^{2}
47 1+74.1T+1.03e5T2 1 + 74.1T + 1.03e5T^{2}
53 1−244.T+1.48e5T2 1 - 244.T + 1.48e5T^{2}
59 1−365.T+2.05e5T2 1 - 365.T + 2.05e5T^{2}
61 1+456.T+2.26e5T2 1 + 456.T + 2.26e5T^{2}
67 1+470.T+3.00e5T2 1 + 470.T + 3.00e5T^{2}
71 1+359.T+3.57e5T2 1 + 359.T + 3.57e5T^{2}
73 1+902.T+3.89e5T2 1 + 902.T + 3.89e5T^{2}
79 1−707.T+4.93e5T2 1 - 707.T + 4.93e5T^{2}
83 1+541.T+5.71e5T2 1 + 541.T + 5.71e5T^{2}
89 1−559.T+7.04e5T2 1 - 559.T + 7.04e5T^{2}
97 1−1.52e3T+9.12e5T2 1 - 1.52e3T + 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

−10.02807710294086412173407651179, −9.369553133563465427284524370280, −8.231248666633591103925035463511, −7.64402503980932719145798507113, −7.06753633136398318874858936043, −5.66008511873508182355819002144, −4.56287687440847775814121742433, −3.21427603445799429985494347293, −1.91627480023023785400789706772, −0.43129763167506888265700262038, 0.43129763167506888265700262038, 1.91627480023023785400789706772, 3.21427603445799429985494347293, 4.56287687440847775814121742433, 5.66008511873508182355819002144, 7.06753633136398318874858936043, 7.64402503980932719145798507113, 8.231248666633591103925035463511, 9.369553133563465427284524370280, 10.02807710294086412173407651179

Graph of the ZZ-function along the critical line