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Properties

Label 2-693-1.1-c3-0-17
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  − 1.94·2-s − 4.22·4-s − 4.10·5-s + 7·7-s + 23.7·8-s + 7.96·10-s + 11·11-s + 71.7·13-s − 13.5·14-s − 12.3·16-s + 79.3·17-s − 159.·19-s + 17.3·20-s − 21.3·22-s − 137.·23-s − 108.·25-s − 139.·26-s − 29.5·28-s + 72.2·29-s + 235.·31-s − 166.·32-s − 154.·34-s − 28.7·35-s − 236.·37-s + 310.·38-s − 97.4·40-s − 147.·41-s + ⋯
L(s)  = 1  − 0.686·2-s − 0.528·4-s − 0.366·5-s + 0.377·7-s + 1.04·8-s + 0.251·10-s + 0.301·11-s + 1.53·13-s − 0.259·14-s − 0.192·16-s + 1.13·17-s − 1.93·19-s + 0.193·20-s − 0.207·22-s − 1.25·23-s − 0.865·25-s − 1.05·26-s − 0.199·28-s + 0.462·29-s + 1.36·31-s − 0.917·32-s − 0.777·34-s − 0.138·35-s − 1.05·37-s + 1.32·38-s − 0.385·40-s − 0.562·41-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 1.1025563491.102556349
L(12)L(\frac12) ≈\approx 1.1025563491.102556349
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+1.94T+8T2 1 + 1.94T + 8T^{2}
5 1+4.10T+125T2 1 + 4.10T + 125T^{2}
13 1−71.7T+2.19e3T2 1 - 71.7T + 2.19e3T^{2}
17 1−79.3T+4.91e3T2 1 - 79.3T + 4.91e3T^{2}
19 1+159.T+6.85e3T2 1 + 159.T + 6.85e3T^{2}
23 1+137.T+1.21e4T2 1 + 137.T + 1.21e4T^{2}
29 1−72.2T+2.43e4T2 1 - 72.2T + 2.43e4T^{2}
31 1−235.T+2.97e4T2 1 - 235.T + 2.97e4T^{2}
37 1+236.T+5.06e4T2 1 + 236.T + 5.06e4T^{2}
41 1+147.T+6.89e4T2 1 + 147.T + 6.89e4T^{2}
43 1+147.T+7.95e4T2 1 + 147.T + 7.95e4T^{2}
47 1−538.T+1.03e5T2 1 - 538.T + 1.03e5T^{2}
53 1−571.T+1.48e5T2 1 - 571.T + 1.48e5T^{2}
59 1−749.T+2.05e5T2 1 - 749.T + 2.05e5T^{2}
61 1+758.T+2.26e5T2 1 + 758.T + 2.26e5T^{2}
67 1+40.4T+3.00e5T2 1 + 40.4T + 3.00e5T^{2}
71 1+131.T+3.57e5T2 1 + 131.T + 3.57e5T^{2}
73 1−238.T+3.89e5T2 1 - 238.T + 3.89e5T^{2}
79 1−588.T+4.93e5T2 1 - 588.T + 4.93e5T^{2}
83 1−481.T+5.71e5T2 1 - 481.T + 5.71e5T^{2}
89 1−1.36e3T+7.04e5T2 1 - 1.36e3T + 7.04e5T^{2}
97 1+667.T+9.12e5T2 1 + 667.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

−10.24095027104907296538447776596, −8.985807989265438122613708101813, −8.354130723611352440966111069775, −7.888930492273827740534429886831, −6.58909122142811947139107185380, −5.63165356491085443003866395235, −4.31472220197792158485871813866, −3.73029386484504597006288422349, −1.86405085266360553966922247707, −0.70130501464473982058510557113, 0.70130501464473982058510557113, 1.86405085266360553966922247707, 3.73029386484504597006288422349, 4.31472220197792158485871813866, 5.63165356491085443003866395235, 6.58909122142811947139107185380, 7.888930492273827740534429886831, 8.354130723611352440966111069775, 8.985807989265438122613708101813, 10.24095027104907296538447776596

Graph of the ZZ-function along the critical line