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Properties

Label 2-693-1.1-c3-0-36
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  − 4.54·2-s + 12.6·4-s − 13.1·5-s + 7·7-s − 21.0·8-s + 59.7·10-s + 11·11-s − 54.8·13-s − 31.7·14-s − 5.62·16-s + 79.9·17-s + 21.2·19-s − 166.·20-s − 49.9·22-s − 119.·23-s + 48.3·25-s + 249.·26-s + 88.3·28-s + 87.4·29-s + 191.·31-s + 193.·32-s − 362.·34-s − 92.1·35-s + 91.6·37-s − 96.5·38-s + 276.·40-s + 60.4·41-s + ⋯
L(s)  = 1  − 1.60·2-s + 1.57·4-s − 1.17·5-s + 0.377·7-s − 0.928·8-s + 1.89·10-s + 0.301·11-s − 1.16·13-s − 0.606·14-s − 0.0878·16-s + 1.14·17-s + 0.256·19-s − 1.85·20-s − 0.484·22-s − 1.08·23-s + 0.386·25-s + 1.87·26-s + 0.596·28-s + 0.560·29-s + 1.10·31-s + 1.06·32-s − 1.83·34-s − 0.445·35-s + 0.407·37-s − 0.412·38-s + 1.09·40-s + 0.230·41-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+4.54T+8T2 1 + 4.54T + 8T^{2}
5 1+13.1T+125T2 1 + 13.1T + 125T^{2}
13 1+54.8T+2.19e3T2 1 + 54.8T + 2.19e3T^{2}
17 1−79.9T+4.91e3T2 1 - 79.9T + 4.91e3T^{2}
19 1−21.2T+6.85e3T2 1 - 21.2T + 6.85e3T^{2}
23 1+119.T+1.21e4T2 1 + 119.T + 1.21e4T^{2}
29 1−87.4T+2.43e4T2 1 - 87.4T + 2.43e4T^{2}
31 1−191.T+2.97e4T2 1 - 191.T + 2.97e4T^{2}
37 1−91.6T+5.06e4T2 1 - 91.6T + 5.06e4T^{2}
41 1−60.4T+6.89e4T2 1 - 60.4T + 6.89e4T^{2}
43 1+213.T+7.95e4T2 1 + 213.T + 7.95e4T^{2}
47 1+417.T+1.03e5T2 1 + 417.T + 1.03e5T^{2}
53 1−414.T+1.48e5T2 1 - 414.T + 1.48e5T^{2}
59 1−358.T+2.05e5T2 1 - 358.T + 2.05e5T^{2}
61 1−515.T+2.26e5T2 1 - 515.T + 2.26e5T^{2}
67 1+107.T+3.00e5T2 1 + 107.T + 3.00e5T^{2}
71 1−711.T+3.57e5T2 1 - 711.T + 3.57e5T^{2}
73 1−131.T+3.89e5T2 1 - 131.T + 3.89e5T^{2}
79 1−10.4T+4.93e5T2 1 - 10.4T + 4.93e5T^{2}
83 1−618.T+5.71e5T2 1 - 618.T + 5.71e5T^{2}
89 1+1.37e3T+7.04e5T2 1 + 1.37e3T + 7.04e5T^{2}
97 1+534.T+9.12e5T2 1 + 534.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

−9.850275722600019857445586262582, −8.573179794888514900965247735255, −7.977559958490869370132602264311, −7.48003209521311523840155229203, −6.54277824958488414116181866851, −5.04500182364427914562724302962, −3.89409295726334963251950373406, −2.50348578530698587819550462608, −1.10613199601176443737102633840, 0, 1.10613199601176443737102633840, 2.50348578530698587819550462608, 3.89409295726334963251950373406, 5.04500182364427914562724302962, 6.54277824958488414116181866851, 7.48003209521311523840155229203, 7.977559958490869370132602264311, 8.573179794888514900965247735255, 9.850275722600019857445586262582

Graph of the ZZ-function along the critical line