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Properties

Label 2-693-1.1-c3-0-39
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 − 18.6·5-s + 7·7-s + 8.80·8-s + 10.4·10-s + 11·11-s + 36.4·13-s − 3.93·14-s + 56.5·16-s − 41.1·17-s − 23.6·19-s + 143.·20-s − 6.17·22-s + 140.·23-s + 224.·25-s − 20.4·26-s − 53.7·28-s − 278.·29-s + 191.·31-s − 102.·32-s + 23.0·34-s − 130.·35-s + 196.·37-s + 13.3·38-s − 164.·40-s + 322.·41-s + ⋯
L(s)  = 1  − 0.198·2-s − 0.960·4-s − 1.67·5-s + 0.377·7-s + 0.389·8-s + 0.331·10-s + 0.301·11-s + 0.777·13-s − 0.0750·14-s + 0.883·16-s − 0.586·17-s − 0.286·19-s + 1.60·20-s − 0.0598·22-s + 1.26·23-s + 1.79·25-s − 0.154·26-s − 0.363·28-s − 1.78·29-s + 1.10·31-s − 0.564·32-s + 0.116·34-s − 0.631·35-s + 0.871·37-s + 0.0568·38-s − 0.650·40-s + 1.22·41-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+18.6T+125T2 1 + 18.6T + 125T^{2}
13 1−36.4T+2.19e3T2 1 - 36.4T + 2.19e3T^{2}
17 1+41.1T+4.91e3T2 1 + 41.1T + 4.91e3T^{2}
19 1+23.6T+6.85e3T2 1 + 23.6T + 6.85e3T^{2}
23 1−140.T+1.21e4T2 1 - 140.T + 1.21e4T^{2}
29 1+278.T+2.43e4T2 1 + 278.T + 2.43e4T^{2}
31 1−191.T+2.97e4T2 1 - 191.T + 2.97e4T^{2}
37 1−196.T+5.06e4T2 1 - 196.T + 5.06e4T^{2}
41 1−322.T+6.89e4T2 1 - 322.T + 6.89e4T^{2}
43 1+3.67T+7.95e4T2 1 + 3.67T + 7.95e4T^{2}
47 1−397.T+1.03e5T2 1 - 397.T + 1.03e5T^{2}
53 1+597.T+1.48e5T2 1 + 597.T + 1.48e5T^{2}
59 1+668.T+2.05e5T2 1 + 668.T + 2.05e5T^{2}
61 1+667.T+2.26e5T2 1 + 667.T + 2.26e5T^{2}
67 1+730.T+3.00e5T2 1 + 730.T + 3.00e5T^{2}
71 1−31.2T+3.57e5T2 1 - 31.2T + 3.57e5T^{2}
73 1+434.T+3.89e5T2 1 + 434.T + 3.89e5T^{2}
79 1+782.T+4.93e5T2 1 + 782.T + 4.93e5T^{2}
83 1−426.T+5.71e5T2 1 - 426.T + 5.71e5T^{2}
89 1−899.T+7.04e5T2 1 - 899.T + 7.04e5T^{2}
97 1+942.T+9.12e5T2 1 + 942.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

−9.291499249620192903756288632822, −8.800852156284164029161057675573, −7.911556582000346082722831442695, −7.37463526735258782032998359880, −6.02916609716696278955398001203, −4.64842969665414758982860636284, −4.19144808374299801080415889668, −3.19288720211848933869566176116, −1.14519559924680600583646109908, 0, 1.14519559924680600583646109908, 3.19288720211848933869566176116, 4.19144808374299801080415889668, 4.64842969665414758982860636284, 6.02916609716696278955398001203, 7.37463526735258782032998359880, 7.911556582000346082722831442695, 8.800852156284164029161057675573, 9.291499249620192903756288632822

Graph of the ZZ-function along the critical line