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Properties

Label 2-693-1.1-c3-0-51
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  − 1.22·2-s − 6.50·4-s + 11.7·5-s − 7·7-s + 17.7·8-s − 14.3·10-s + 11·11-s − 25.5·13-s + 8.56·14-s + 30.3·16-s − 16.3·17-s − 126.·19-s − 76.1·20-s − 13.4·22-s + 94.6·23-s + 12.2·25-s + 31.2·26-s + 45.5·28-s + 160.·29-s − 12.1·31-s − 179.·32-s + 19.9·34-s − 82.0·35-s + 436.·37-s + 154.·38-s + 207.·40-s − 264.·41-s + ⋯
L(s)  = 1  − 0.432·2-s − 0.812·4-s + 1.04·5-s − 0.377·7-s + 0.784·8-s − 0.453·10-s + 0.301·11-s − 0.545·13-s + 0.163·14-s + 0.473·16-s − 0.232·17-s − 1.52·19-s − 0.851·20-s − 0.130·22-s + 0.858·23-s + 0.0979·25-s + 0.235·26-s + 0.307·28-s + 1.02·29-s − 0.0701·31-s − 0.989·32-s + 0.100·34-s − 0.396·35-s + 1.94·37-s + 0.658·38-s + 0.821·40-s − 1.00·41-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+1.22T+8T2 1 + 1.22T + 8T^{2}
5 1−11.7T+125T2 1 - 11.7T + 125T^{2}
13 1+25.5T+2.19e3T2 1 + 25.5T + 2.19e3T^{2}
17 1+16.3T+4.91e3T2 1 + 16.3T + 4.91e3T^{2}
19 1+126.T+6.85e3T2 1 + 126.T + 6.85e3T^{2}
23 1−94.6T+1.21e4T2 1 - 94.6T + 1.21e4T^{2}
29 1−160.T+2.43e4T2 1 - 160.T + 2.43e4T^{2}
31 1+12.1T+2.97e4T2 1 + 12.1T + 2.97e4T^{2}
37 1−436.T+5.06e4T2 1 - 436.T + 5.06e4T^{2}
41 1+264.T+6.89e4T2 1 + 264.T + 6.89e4T^{2}
43 1+171.T+7.95e4T2 1 + 171.T + 7.95e4T^{2}
47 1+535.T+1.03e5T2 1 + 535.T + 1.03e5T^{2}
53 1−514.T+1.48e5T2 1 - 514.T + 1.48e5T^{2}
59 1+607.T+2.05e5T2 1 + 607.T + 2.05e5T^{2}
61 1−47.8T+2.26e5T2 1 - 47.8T + 2.26e5T^{2}
67 1−1.05e3T+3.00e5T2 1 - 1.05e3T + 3.00e5T^{2}
71 1+783.T+3.57e5T2 1 + 783.T + 3.57e5T^{2}
73 1+1.10e3T+3.89e5T2 1 + 1.10e3T + 3.89e5T^{2}
79 1+194.T+4.93e5T2 1 + 194.T + 4.93e5T^{2}
83 1+489.T+5.71e5T2 1 + 489.T + 5.71e5T^{2}
89 1+466.T+7.04e5T2 1 + 466.T + 7.04e5T^{2}
97 1+429.T+9.12e5T2 1 + 429.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

−9.710544162767554588522755751896, −8.878194449188127204568215175168, −8.182434701725897214312592908330, −6.91397090733053461628800919437, −6.10929774538131927725444006893, −5.03141479785759324148363275160, −4.17169640590165057727629389742, −2.69563789644643788545369292425, −1.43396746574585920656150309750, 0, 1.43396746574585920656150309750, 2.69563789644643788545369292425, 4.17169640590165057727629389742, 5.03141479785759324148363275160, 6.10929774538131927725444006893, 6.91397090733053461628800919437, 8.182434701725897214312592908330, 8.878194449188127204568215175168, 9.710544162767554588522755751896

Graph of the ZZ-function along the critical line