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Properties

Label 2-693-1.1-c3-0-47
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  − 0.948·2-s − 7.10·4-s + 5.36·5-s − 7·7-s + 14.3·8-s − 5.09·10-s − 11·11-s − 42.9·13-s + 6.64·14-s + 43.2·16-s + 60.8·17-s + 140.·19-s − 38.1·20-s + 10.4·22-s + 91.3·23-s − 96.1·25-s + 40.7·26-s + 49.7·28-s − 260.·29-s − 259.·31-s − 155.·32-s − 57.7·34-s − 37.5·35-s + 359.·37-s − 133.·38-s + 76.8·40-s + 320.·41-s + ⋯
L(s)  = 1  − 0.335·2-s − 0.887·4-s + 0.480·5-s − 0.377·7-s + 0.633·8-s − 0.161·10-s − 0.301·11-s − 0.916·13-s + 0.126·14-s + 0.675·16-s + 0.868·17-s + 1.69·19-s − 0.426·20-s + 0.101·22-s + 0.828·23-s − 0.769·25-s + 0.307·26-s + 0.335·28-s − 1.67·29-s − 1.50·31-s − 0.859·32-s − 0.291·34-s − 0.181·35-s + 1.59·37-s − 0.569·38-s + 0.303·40-s + 1.21·41-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+0.948T+8T2 1 + 0.948T + 8T^{2}
5 1−5.36T+125T2 1 - 5.36T + 125T^{2}
13 1+42.9T+2.19e3T2 1 + 42.9T + 2.19e3T^{2}
17 1−60.8T+4.91e3T2 1 - 60.8T + 4.91e3T^{2}
19 1−140.T+6.85e3T2 1 - 140.T + 6.85e3T^{2}
23 1−91.3T+1.21e4T2 1 - 91.3T + 1.21e4T^{2}
29 1+260.T+2.43e4T2 1 + 260.T + 2.43e4T^{2}
31 1+259.T+2.97e4T2 1 + 259.T + 2.97e4T^{2}
37 1−359.T+5.06e4T2 1 - 359.T + 5.06e4T^{2}
41 1−320.T+6.89e4T2 1 - 320.T + 6.89e4T^{2}
43 1+92.3T+7.95e4T2 1 + 92.3T + 7.95e4T^{2}
47 1+67.4T+1.03e5T2 1 + 67.4T + 1.03e5T^{2}
53 1−246.T+1.48e5T2 1 - 246.T + 1.48e5T^{2}
59 1−475.T+2.05e5T2 1 - 475.T + 2.05e5T^{2}
61 1+799.T+2.26e5T2 1 + 799.T + 2.26e5T^{2}
67 1+725.T+3.00e5T2 1 + 725.T + 3.00e5T^{2}
71 1+544.T+3.57e5T2 1 + 544.T + 3.57e5T^{2}
73 1+580.T+3.89e5T2 1 + 580.T + 3.89e5T^{2}
79 1+402.T+4.93e5T2 1 + 402.T + 4.93e5T^{2}
83 1−1.10e3T+5.71e5T2 1 - 1.10e3T + 5.71e5T^{2}
89 1+1.25e3T+7.04e5T2 1 + 1.25e3T + 7.04e5T^{2}
97 1+9.99e2T+9.12e5T2 1 + 9.99e2T + 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

−9.479264504763567314750885243206, −9.184441188353012851282104559707, −7.68327679839171147352884729232, −7.42069251850984945151730663946, −5.72566311713299537455326859646, −5.27849320457487709593338216840, −4.01275152710589195220119942013, −2.89011072785254005961638972133, −1.34468543103861334978274148012, 0, 1.34468543103861334978274148012, 2.89011072785254005961638972133, 4.01275152710589195220119942013, 5.27849320457487709593338216840, 5.72566311713299537455326859646, 7.42069251850984945151730663946, 7.68327679839171147352884729232, 9.184441188353012851282104559707, 9.479264504763567314750885243206

Graph of the ZZ-function along the critical line