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Properties

Label 2-693-1.1-c3-0-19
Degree 22
Conductor 693693
Sign 11
Analytic cond. 40.888340.8883
Root an. cond. 6.394396.39439
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·4-s − 5-s − 7·7-s − 24·8-s − 2·10-s + 11·11-s + 7·13-s − 14·14-s − 16·16-s + 14·17-s − 45·19-s + 4·20-s + 22·22-s + 88·23-s − 124·25-s + 14·26-s + 28·28-s + 69·29-s + 22·31-s + 160·32-s + 28·34-s + 7·35-s + 57·37-s − 90·38-s + 24·40-s + 380·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.0894·5-s − 0.377·7-s − 1.06·8-s − 0.0632·10-s + 0.301·11-s + 0.149·13-s − 0.267·14-s − 1/4·16-s + 0.199·17-s − 0.543·19-s + 0.0447·20-s + 0.213·22-s + 0.797·23-s − 0.991·25-s + 0.105·26-s + 0.188·28-s + 0.441·29-s + 0.127·31-s + 0.883·32-s + 0.141·34-s + 0.0338·35-s + 0.253·37-s − 0.384·38-s + 0.0948·40-s + 1.44·41-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: 11
Analytic conductor: 40.888340.8883
Root analytic conductor: 6.394396.39439
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 693, ( :3/2), 1)(2,\ 693,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.9911445521.991144552
L(12)L(\frac12) ≈\approx 1.9911445521.991144552
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+pT 1 + p T
11 1−pT 1 - p T
good2 1−pT+p3T2 1 - p T + p^{3} T^{2}
5 1+T+p3T2 1 + T + p^{3} T^{2}
13 1−7T+p3T2 1 - 7 T + p^{3} T^{2}
17 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
19 1+45T+p3T2 1 + 45 T + p^{3} T^{2}
23 1−88T+p3T2 1 - 88 T + p^{3} T^{2}
29 1−69T+p3T2 1 - 69 T + p^{3} T^{2}
31 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
37 1−57T+p3T2 1 - 57 T + p^{3} T^{2}
41 1−380T+p3T2 1 - 380 T + p^{3} T^{2}
43 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
47 1−385T+p3T2 1 - 385 T + p^{3} T^{2}
53 1−672T+p3T2 1 - 672 T + p^{3} T^{2}
59 1−469T+p3T2 1 - 469 T + p^{3} T^{2}
61 1+342T+p3T2 1 + 342 T + p^{3} T^{2}
67 1+139T+p3T2 1 + 139 T + p^{3} T^{2}
71 1+132T+p3T2 1 + 132 T + p^{3} T^{2}
73 1−145T+p3T2 1 - 145 T + p^{3} T^{2}
79 1−1244T+p3T2 1 - 1244 T + p^{3} T^{2}
83 1+522T+p3T2 1 + 522 T + p^{3} T^{2}
89 1+822T+p3T2 1 + 822 T + p^{3} T^{2}
97 1−272T+p3T2 1 - 272 T + p^{3} T^{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

−9.987994068731365345071473404866, −9.186896069311863973039843929473, −8.473894102148616565102028162203, −7.34223425182523230787220981400, −6.25905327476354122980015967423, −5.53730142149248904993793084082, −4.41858659939594583211532429344, −3.70021117167718319732584525304, −2.55313846555331212411806747010, −0.73257860969446132913644639821, 0.73257860969446132913644639821, 2.55313846555331212411806747010, 3.70021117167718319732584525304, 4.41858659939594583211532429344, 5.53730142149248904993793084082, 6.25905327476354122980015967423, 7.34223425182523230787220981400, 8.473894102148616565102028162203, 9.186896069311863973039843929473, 9.987994068731365345071473404866

Graph of the ZZ-function along the critical line