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Properties

Label 2-812-203.202-c0-0-0
Degree 22
Conductor 812812
Sign 11
Analytic cond. 0.4052400.405240
Root an. cond. 0.6365850.636585
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·3-s − 7-s + 1.99·9-s + 1.73·19-s + 1.73·21-s + 23-s + 25-s − 1.73·27-s − 29-s + 1.73·41-s + 1.73·47-s + 49-s − 53-s − 2.99·57-s − 1.99·63-s + 67-s − 1.73·69-s − 71-s − 1.73·73-s − 1.73·75-s + 0.999·81-s + 1.73·87-s − 1.73·89-s + 1.73·97-s − 1.73·101-s − 107-s + 109-s + ⋯
L(s)  = 1  − 1.73·3-s − 7-s + 1.99·9-s + 1.73·19-s + 1.73·21-s + 23-s + 25-s − 1.73·27-s − 29-s + 1.73·41-s + 1.73·47-s + 49-s − 53-s − 2.99·57-s − 1.99·63-s + 67-s − 1.73·69-s − 71-s − 1.73·73-s − 1.73·75-s + 0.999·81-s + 1.73·87-s − 1.73·89-s + 1.73·97-s − 1.73·101-s − 107-s + 109-s + ⋯

Functional equation

Λ(s)=(812s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(812s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 812812    =    22⋅7⋅292^{2} \cdot 7 \cdot 29
Sign: 11
Analytic conductor: 0.4052400.405240
Root analytic conductor: 0.6365850.636585
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ812(405,⋅)\chi_{812} (405, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 812, ( :0), 1)(2,\ 812,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.52056913510.5205691351
L(12)L(\frac12) ≈\approx 0.52056913510.5205691351
L(1)L(1) 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)
bad2 1 1
7 1+T 1 + T
29 1+T 1 + T
good3 1+1.73T+T2 1 + 1.73T + T^{2}
5 1−T2 1 - T^{2}
11 1−T2 1 - T^{2}
13 1−T2 1 - T^{2}
17 1+T2 1 + T^{2}
19 1−1.73T+T2 1 - 1.73T + T^{2}
23 1−T+T2 1 - T + T^{2}
31 1+T2 1 + T^{2}
37 1−T2 1 - T^{2}
41 1−1.73T+T2 1 - 1.73T + T^{2}
43 1−T2 1 - T^{2}
47 1−1.73T+T2 1 - 1.73T + T^{2}
53 1+T+T2 1 + T + T^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1−T+T2 1 - T + T^{2}
71 1+T+T2 1 + T + T^{2}
73 1+1.73T+T2 1 + 1.73T + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+1.73T+T2 1 + 1.73T + T^{2}
97 1−1.73T+T2 1 - 1.73T + 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

−10.64412267982860464758153320361, −9.733817669010682121482022605962, −9.096389976472646023040836909190, −7.42706365991508106006628374285, −6.93561321851813590221331096498, −5.91043855455644174829468058397, −5.39650767086687713977134317255, −4.32668210146249067638925109065, −3.04287168102832605721011115747, −0.982292206454670002103035977039, 0.982292206454670002103035977039, 3.04287168102832605721011115747, 4.32668210146249067638925109065, 5.39650767086687713977134317255, 5.91043855455644174829468058397, 6.93561321851813590221331096498, 7.42706365991508106006628374285, 9.096389976472646023040836909190, 9.733817669010682121482022605962, 10.64412267982860464758153320361

Graph of the ZZ-function along the critical line