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Properties

Label 2-693-1.1-c3-0-49
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  + 0.561·2-s − 7.68·4-s − 6.68·5-s + 7·7-s − 8.80·8-s − 3.75·10-s + 11·11-s + 14.3·13-s + 3.93·14-s + 56.5·16-s + 47.7·17-s + 11.9·19-s + 51.3·20-s + 6.17·22-s + 44.4·23-s − 80.3·25-s + 8.03·26-s − 53.7·28-s + 139.·29-s − 208.·31-s + 102.·32-s + 26.8·34-s − 46.7·35-s − 253.·37-s + 6.69·38-s + 58.8·40-s + 156.·41-s + ⋯
L(s)  = 1  + 0.198·2-s − 0.960·4-s − 0.597·5-s + 0.377·7-s − 0.389·8-s − 0.118·10-s + 0.301·11-s + 0.305·13-s + 0.0750·14-s + 0.883·16-s + 0.681·17-s + 0.143·19-s + 0.574·20-s + 0.0598·22-s + 0.403·23-s − 0.642·25-s + 0.0605·26-s − 0.363·28-s + 0.893·29-s − 1.20·31-s + 0.564·32-s + 0.135·34-s − 0.225·35-s − 1.12·37-s + 0.0285·38-s + 0.232·40-s + 0.595·41-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 1−0.561T+8T2 1 - 0.561T + 8T^{2}
5 1+6.68T+125T2 1 + 6.68T + 125T^{2}
13 1−14.3T+2.19e3T2 1 - 14.3T + 2.19e3T^{2}
17 1−47.7T+4.91e3T2 1 - 47.7T + 4.91e3T^{2}
19 1−11.9T+6.85e3T2 1 - 11.9T + 6.85e3T^{2}
23 1−44.4T+1.21e4T2 1 - 44.4T + 1.21e4T^{2}
29 1−139.T+2.43e4T2 1 - 139.T + 2.43e4T^{2}
31 1+208.T+2.97e4T2 1 + 208.T + 2.97e4T^{2}
37 1+253.T+5.06e4T2 1 + 253.T + 5.06e4T^{2}
41 1−156.T+6.89e4T2 1 - 156.T + 6.89e4T^{2}
43 1+263.T+7.95e4T2 1 + 263.T + 7.95e4T^{2}
47 1+386.T+1.03e5T2 1 + 386.T + 1.03e5T^{2}
53 1−36.5T+1.48e5T2 1 - 36.5T + 1.48e5T^{2}
59 1−114.T+2.05e5T2 1 - 114.T + 2.05e5T^{2}
61 1+53.0T+2.26e5T2 1 + 53.0T + 2.26e5T^{2}
67 1−132.T+3.00e5T2 1 - 132.T + 3.00e5T^{2}
71 1+583.T+3.57e5T2 1 + 583.T + 3.57e5T^{2}
73 1+817.T+3.89e5T2 1 + 817.T + 3.89e5T^{2}
79 1+369.T+4.93e5T2 1 + 369.T + 4.93e5T^{2}
83 1−69.1T+5.71e5T2 1 - 69.1T + 5.71e5T^{2}
89 1+467.T+7.04e5T2 1 + 467.T + 7.04e5T^{2}
97 1+1.17e3T+9.12e5T2 1 + 1.17e3T + 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

−9.579425032450930458840847296578, −8.685274798969743728111074218185, −8.053926276143849475536081811005, −7.09458888031229806519488101139, −5.82932443745915256131683548196, −4.97061842491356212840182660399, −4.03273550061096967927586777432, −3.21851181291526248165313877429, −1.37622371514551312910643013372, 0, 1.37622371514551312910643013372, 3.21851181291526248165313877429, 4.03273550061096967927586777432, 4.97061842491356212840182660399, 5.82932443745915256131683548196, 7.09458888031229806519488101139, 8.053926276143849475536081811005, 8.685274798969743728111074218185, 9.579425032450930458840847296578

Graph of the ZZ-function along the critical line