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Properties

Label 2-693-1.1-c3-0-31
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  − 3.85·2-s + 6.86·4-s − 18.0·5-s − 7·7-s + 4.37·8-s + 69.5·10-s − 11·11-s + 16.8·13-s + 26.9·14-s − 71.7·16-s − 76.0·17-s + 11.8·19-s − 123.·20-s + 42.4·22-s + 175.·23-s + 200.·25-s − 64.9·26-s − 48.0·28-s + 219.·29-s − 214.·31-s + 241.·32-s + 293.·34-s + 126.·35-s + 272.·37-s − 45.8·38-s − 78.9·40-s − 204.·41-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.858·4-s − 1.61·5-s − 0.377·7-s + 0.193·8-s + 2.19·10-s − 0.301·11-s + 0.359·13-s + 0.515·14-s − 1.12·16-s − 1.08·17-s + 0.143·19-s − 1.38·20-s + 0.410·22-s + 1.59·23-s + 1.60·25-s − 0.489·26-s − 0.324·28-s + 1.40·29-s − 1.24·31-s + 1.33·32-s + 1.47·34-s + 0.609·35-s + 1.21·37-s − 0.195·38-s − 0.312·40-s − 0.780·41-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+3.85T+8T2 1 + 3.85T + 8T^{2}
5 1+18.0T+125T2 1 + 18.0T + 125T^{2}
13 1−16.8T+2.19e3T2 1 - 16.8T + 2.19e3T^{2}
17 1+76.0T+4.91e3T2 1 + 76.0T + 4.91e3T^{2}
19 1−11.8T+6.85e3T2 1 - 11.8T + 6.85e3T^{2}
23 1−175.T+1.21e4T2 1 - 175.T + 1.21e4T^{2}
29 1−219.T+2.43e4T2 1 - 219.T + 2.43e4T^{2}
31 1+214.T+2.97e4T2 1 + 214.T + 2.97e4T^{2}
37 1−272.T+5.06e4T2 1 - 272.T + 5.06e4T^{2}
41 1+204.T+6.89e4T2 1 + 204.T + 6.89e4T^{2}
43 1−406.T+7.95e4T2 1 - 406.T + 7.95e4T^{2}
47 1+178.T+1.03e5T2 1 + 178.T + 1.03e5T^{2}
53 1−238.T+1.48e5T2 1 - 238.T + 1.48e5T^{2}
59 1−584.T+2.05e5T2 1 - 584.T + 2.05e5T^{2}
61 1+910.T+2.26e5T2 1 + 910.T + 2.26e5T^{2}
67 1+387.T+3.00e5T2 1 + 387.T + 3.00e5T^{2}
71 1−668.T+3.57e5T2 1 - 668.T + 3.57e5T^{2}
73 1−390.T+3.89e5T2 1 - 390.T + 3.89e5T^{2}
79 1+250.T+4.93e5T2 1 + 250.T + 4.93e5T^{2}
83 1−54.2T+5.71e5T2 1 - 54.2T + 5.71e5T^{2}
89 1−906.T+7.04e5T2 1 - 906.T + 7.04e5T^{2}
97 1−618.T+9.12e5T2 1 - 618.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

−9.368298233270626770570725351583, −8.766854187541899880315934509307, −8.027205823984859827455744637411, −7.29640448502911741119679744593, −6.59266412295047314007437206080, −4.88603513088611670629321519876, −3.95552492277847087508546426587, −2.71817517107884976265953999878, −0.978046534140947832490399795836, 0, 0.978046534140947832490399795836, 2.71817517107884976265953999878, 3.95552492277847087508546426587, 4.88603513088611670629321519876, 6.59266412295047314007437206080, 7.29640448502911741119679744593, 8.027205823984859827455744637411, 8.766854187541899880315934509307, 9.368298233270626770570725351583

Graph of the ZZ-function along the critical line