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Properties

Label 2-693-1.1-c3-0-37
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  − 5.62·2-s + 23.6·4-s − 7.12·5-s − 7·7-s − 87.8·8-s + 40.0·10-s + 11·11-s − 28.1·13-s + 39.3·14-s + 305.·16-s − 53.4·17-s + 154.·19-s − 168.·20-s − 61.8·22-s + 44.4·23-s − 74.3·25-s + 158.·26-s − 165.·28-s + 154.·29-s + 63.2·31-s − 1.01e3·32-s + 300.·34-s + 49.8·35-s − 137.·37-s − 866.·38-s + 625.·40-s − 442.·41-s + ⋯
L(s)  = 1  − 1.98·2-s + 2.95·4-s − 0.636·5-s − 0.377·7-s − 3.88·8-s + 1.26·10-s + 0.301·11-s − 0.600·13-s + 0.751·14-s + 4.76·16-s − 0.762·17-s + 1.86·19-s − 1.88·20-s − 0.599·22-s + 0.403·23-s − 0.594·25-s + 1.19·26-s − 1.11·28-s + 0.988·29-s + 0.366·31-s − 5.59·32-s + 1.51·34-s + 0.240·35-s − 0.611·37-s − 3.69·38-s + 2.47·40-s − 1.68·41-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+5.62T+8T2 1 + 5.62T + 8T^{2}
5 1+7.12T+125T2 1 + 7.12T + 125T^{2}
13 1+28.1T+2.19e3T2 1 + 28.1T + 2.19e3T^{2}
17 1+53.4T+4.91e3T2 1 + 53.4T + 4.91e3T^{2}
19 1−154.T+6.85e3T2 1 - 154.T + 6.85e3T^{2}
23 1−44.4T+1.21e4T2 1 - 44.4T + 1.21e4T^{2}
29 1−154.T+2.43e4T2 1 - 154.T + 2.43e4T^{2}
31 1−63.2T+2.97e4T2 1 - 63.2T + 2.97e4T^{2}
37 1+137.T+5.06e4T2 1 + 137.T + 5.06e4T^{2}
41 1+442.T+6.89e4T2 1 + 442.T + 6.89e4T^{2}
43 1+48.2T+7.95e4T2 1 + 48.2T + 7.95e4T^{2}
47 1−122.T+1.03e5T2 1 - 122.T + 1.03e5T^{2}
53 1−298.T+1.48e5T2 1 - 298.T + 1.48e5T^{2}
59 1−694.T+2.05e5T2 1 - 694.T + 2.05e5T^{2}
61 1+7.85T+2.26e5T2 1 + 7.85T + 2.26e5T^{2}
67 1−467.T+3.00e5T2 1 - 467.T + 3.00e5T^{2}
71 1−984.T+3.57e5T2 1 - 984.T + 3.57e5T^{2}
73 1+128.T+3.89e5T2 1 + 128.T + 3.89e5T^{2}
79 1+1.33e3T+4.93e5T2 1 + 1.33e3T + 4.93e5T^{2}
83 1−984.T+5.71e5T2 1 - 984.T + 5.71e5T^{2}
89 1+918.T+7.04e5T2 1 + 918.T + 7.04e5T^{2}
97 1+1.53e3T+9.12e5T2 1 + 1.53e3T + 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

−9.687046461794374040974517545338, −8.761930441574296621735906105898, −8.085987287474218385717142510177, −7.14911389672617409198250646391, −6.71876740315079389585767362723, −5.39809819897996569740280924410, −3.52134576320278526122207731924, −2.49049010650185843913023122691, −1.11533686454598848017752590049, 0, 1.11533686454598848017752590049, 2.49049010650185843913023122691, 3.52134576320278526122207731924, 5.39809819897996569740280924410, 6.71876740315079389585767362723, 7.14911389672617409198250646391, 8.085987287474218385717142510177, 8.761930441574296621735906105898, 9.687046461794374040974517545338

Graph of the ZZ-function along the critical line