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Properties

Label 2-693-1.1-c3-0-23
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  − 3·2-s + 4-s + 14·5-s − 7·7-s + 21·8-s − 42·10-s + 11·11-s + 2·13-s + 21·14-s − 71·16-s + 74·17-s + 14·20-s − 33·22-s + 148·23-s + 71·25-s − 6·26-s − 7·28-s − 26·29-s + 112·31-s + 45·32-s − 222·34-s − 98·35-s − 98·37-s + 294·40-s + 10·41-s + 208·43-s + 11·44-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s + 1.25·5-s − 0.377·7-s + 0.928·8-s − 1.32·10-s + 0.301·11-s + 0.0426·13-s + 0.400·14-s − 1.10·16-s + 1.05·17-s + 0.156·20-s − 0.319·22-s + 1.34·23-s + 0.567·25-s − 0.0452·26-s − 0.0472·28-s − 0.166·29-s + 0.648·31-s + 0.248·32-s − 1.11·34-s − 0.473·35-s − 0.435·37-s + 1.16·40-s + 0.0380·41-s + 0.737·43-s + 0.0376·44-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 1.3815336401.381533640
L(12)L(\frac12) ≈\approx 1.3815336401.381533640
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+3T+p3T2 1 + 3 T + p^{3} T^{2}
5 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
13 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
17 1−74T+p3T2 1 - 74 T + p^{3} T^{2}
19 1+p3T2 1 + p^{3} T^{2}
23 1−148T+p3T2 1 - 148 T + p^{3} T^{2}
29 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
31 1−112T+p3T2 1 - 112 T + p^{3} T^{2}
37 1+98T+p3T2 1 + 98 T + p^{3} T^{2}
41 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
43 1−208T+p3T2 1 - 208 T + p^{3} T^{2}
47 1+460T+p3T2 1 + 460 T + p^{3} T^{2}
53 1+258T+p3T2 1 + 258 T + p^{3} T^{2}
59 1−204T+p3T2 1 - 204 T + p^{3} T^{2}
61 1−178T+p3T2 1 - 178 T + p^{3} T^{2}
67 1+924T+p3T2 1 + 924 T + p^{3} T^{2}
71 1−748T+p3T2 1 - 748 T + p^{3} T^{2}
73 1+230T+p3T2 1 + 230 T + p^{3} T^{2}
79 1+456T+p3T2 1 + 456 T + p^{3} T^{2}
83 1−228T+p3T2 1 - 228 T + p^{3} T^{2}
89 1−198T+p3T2 1 - 198 T + p^{3} T^{2}
97 1−562T+p3T2 1 - 562 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

−9.886398178248379544860494558359, −9.329199835206800537243620382934, −8.605538940027315713883353534933, −7.57620374228745966013744538783, −6.65841078082897294538706593515, −5.69459977783263275639729743975, −4.70921456100148149312023808814, −3.20298106283103105230625728271, −1.83983440378421350374546410015, −0.841711287666450173254551209289, 0.841711287666450173254551209289, 1.83983440378421350374546410015, 3.20298106283103105230625728271, 4.70921456100148149312023808814, 5.69459977783263275639729743975, 6.65841078082897294538706593515, 7.57620374228745966013744538783, 8.605538940027315713883353534933, 9.329199835206800537243620382934, 9.886398178248379544860494558359

Graph of the ZZ-function along the critical line