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Properties

Label 2-693-1.1-c3-0-44
Degree 22
Conductor 693693
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.315·2-s − 7.90·4-s − 5.90·5-s − 7·7-s + 5.02·8-s + 1.86·10-s + 11·11-s + 2.91·13-s + 2.21·14-s + 61.6·16-s + 60.0·17-s + 69.8·19-s + 46.6·20-s − 3.47·22-s + 120.·23-s − 90.0·25-s − 0.919·26-s + 55.3·28-s − 174.·29-s + 44.5·31-s − 59.6·32-s − 18.9·34-s + 41.3·35-s − 271.·37-s − 22.0·38-s − 29.6·40-s − 355.·41-s + ⋯
L(s)  = 1  − 0.111·2-s − 0.987·4-s − 0.528·5-s − 0.377·7-s + 0.221·8-s + 0.0590·10-s + 0.301·11-s + 0.0621·13-s + 0.0422·14-s + 0.962·16-s + 0.856·17-s + 0.843·19-s + 0.521·20-s − 0.0336·22-s + 1.09·23-s − 0.720·25-s − 0.00693·26-s + 0.373·28-s − 1.11·29-s + 0.258·31-s − 0.329·32-s − 0.0956·34-s + 0.199·35-s − 1.20·37-s − 0.0942·38-s − 0.117·40-s − 1.35·41-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: −1-1
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: 11
Selberg data: (2, 693, ( :3/2), −1)(2,\ 693,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+0.315T+8T2 1 + 0.315T + 8T^{2}
5 1+5.90T+125T2 1 + 5.90T + 125T^{2}
13 1−2.91T+2.19e3T2 1 - 2.91T + 2.19e3T^{2}
17 1−60.0T+4.91e3T2 1 - 60.0T + 4.91e3T^{2}
19 1−69.8T+6.85e3T2 1 - 69.8T + 6.85e3T^{2}
23 1−120.T+1.21e4T2 1 - 120.T + 1.21e4T^{2}
29 1+174.T+2.43e4T2 1 + 174.T + 2.43e4T^{2}
31 1−44.5T+2.97e4T2 1 - 44.5T + 2.97e4T^{2}
37 1+271.T+5.06e4T2 1 + 271.T + 5.06e4T^{2}
41 1+355.T+6.89e4T2 1 + 355.T + 6.89e4T^{2}
43 1−545.T+7.95e4T2 1 - 545.T + 7.95e4T^{2}
47 1−413.T+1.03e5T2 1 - 413.T + 1.03e5T^{2}
53 1+709.T+1.48e5T2 1 + 709.T + 1.48e5T^{2}
59 1+358.T+2.05e5T2 1 + 358.T + 2.05e5T^{2}
61 1+574.T+2.26e5T2 1 + 574.T + 2.26e5T^{2}
67 1−457.T+3.00e5T2 1 - 457.T + 3.00e5T^{2}
71 1+87.8T+3.57e5T2 1 + 87.8T + 3.57e5T^{2}
73 1−403.T+3.89e5T2 1 - 403.T + 3.89e5T^{2}
79 1+195.T+4.93e5T2 1 + 195.T + 4.93e5T^{2}
83 1+1.40e3T+5.71e5T2 1 + 1.40e3T + 5.71e5T^{2}
89 1+1.25e3T+7.04e5T2 1 + 1.25e3T + 7.04e5T^{2}
97 1−1.77e3T+9.12e5T2 1 - 1.77e3T + 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

−9.483927545814833006735917263696, −8.935989150057316260350863846174, −7.87963153030738317908763155338, −7.23726960336147070992319373448, −5.88678598443091446871308240639, −5.03746742517129207158523825681, −3.93539683846517281856013301383, −3.18509388841564434052980060059, −1.25960653393990835476016972010, 0, 1.25960653393990835476016972010, 3.18509388841564434052980060059, 3.93539683846517281856013301383, 5.03746742517129207158523825681, 5.88678598443091446871308240639, 7.23726960336147070992319373448, 7.87963153030738317908763155338, 8.935989150057316260350863846174, 9.483927545814833006735917263696

Graph of the ZZ-function along the critical line