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Properties

Label 2-1083-1.1-c5-0-22
Degree 22
Conductor 10831083
Sign 11
Analytic cond. 173.695173.695
Root an. cond. 13.179313.1793
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11·2-s − 9·3-s + 89·4-s + 6·5-s + 99·6-s − 176·7-s − 627·8-s + 81·9-s − 66·10-s − 496·11-s − 801·12-s + 178·13-s + 1.93e3·14-s − 54·15-s + 4.04e3·16-s + 202·17-s − 891·18-s + 534·20-s + 1.58e3·21-s + 5.45e3·22-s + 4.39e3·23-s + 5.64e3·24-s − 3.08e3·25-s − 1.95e3·26-s − 729·27-s − 1.56e4·28-s + 5.90e3·29-s + ⋯
L(s)  = 1  − 1.94·2-s − 0.577·3-s + 2.78·4-s + 0.107·5-s + 1.12·6-s − 1.35·7-s − 3.46·8-s + 1/3·9-s − 0.208·10-s − 1.23·11-s − 1.60·12-s + 0.292·13-s + 2.63·14-s − 0.0619·15-s + 3.95·16-s + 0.169·17-s − 0.648·18-s + 0.298·20-s + 0.783·21-s + 2.40·22-s + 1.73·23-s + 1.99·24-s − 0.988·25-s − 0.568·26-s − 0.192·27-s − 3.77·28-s + 1.30·29-s + ⋯

Functional equation

Λ(s)=(1083s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(1083s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10831083    =    3⋅1923 \cdot 19^{2}
Sign: 11
Analytic conductor: 173.695173.695
Root analytic conductor: 13.179313.1793
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1083, ( :5/2), 1)(2,\ 1083,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.19156382840.1915638284
L(12)L(\frac12) ≈\approx 0.19156382840.1915638284
L(72)L(\frac{7}{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+p2T 1 + p^{2} T
19 1 1
good2 1+11T+p5T2 1 + 11 T + p^{5} T^{2}
5 1−6T+p5T2 1 - 6 T + p^{5} T^{2}
7 1+176T+p5T2 1 + 176 T + p^{5} T^{2}
11 1+496T+p5T2 1 + 496 T + p^{5} T^{2}
13 1−178T+p5T2 1 - 178 T + p^{5} T^{2}
17 1−202T+p5T2 1 - 202 T + p^{5} T^{2}
23 1−4396T+p5T2 1 - 4396 T + p^{5} T^{2}
29 1−5902T+p5T2 1 - 5902 T + p^{5} T^{2}
31 1+5760T+p5T2 1 + 5760 T + p^{5} T^{2}
37 1−3906T+p5T2 1 - 3906 T + p^{5} T^{2}
41 1+15774T+p5T2 1 + 15774 T + p^{5} T^{2}
43 1+7492T+p5T2 1 + 7492 T + p^{5} T^{2}
47 1+7452T+p5T2 1 + 7452 T + p^{5} T^{2}
53 1−29014T+p5T2 1 - 29014 T + p^{5} T^{2}
59 1+13604T+p5T2 1 + 13604 T + p^{5} T^{2}
61 1+12466T+p5T2 1 + 12466 T + p^{5} T^{2}
67 1+43436T+p5T2 1 + 43436 T + p^{5} T^{2}
71 1+28800T+p5T2 1 + 28800 T + p^{5} T^{2}
73 1−80746T+p5T2 1 - 80746 T + p^{5} T^{2}
79 1+76456T+p5T2 1 + 76456 T + p^{5} T^{2}
83 1+56880T+p5T2 1 + 56880 T + p^{5} T^{2}
89 1−103266T+p5T2 1 - 103266 T + p^{5} T^{2}
97 1+82490T+p5T2 1 + 82490 T + p^{5} 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

−9.230837949553101819544264054126, −8.479332976153336234821789124284, −7.53777529137282696099431555355, −6.85642139949342265882593230155, −6.18159206849484436597535026360, −5.28557373129483663195806586323, −3.33458507810404999349395751418, −2.58319887526859354826789805826, −1.32778140209485255123253892036, −0.27555143314703734825203014013, 0.27555143314703734825203014013, 1.32778140209485255123253892036, 2.58319887526859354826789805826, 3.33458507810404999349395751418, 5.28557373129483663195806586323, 6.18159206849484436597535026360, 6.85642139949342265882593230155, 7.53777529137282696099431555355, 8.479332976153336234821789124284, 9.230837949553101819544264054126

Graph of the ZZ-function along the critical line