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Properties

Label 2-1083-1.1-c5-0-28
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  + 6·2-s − 9·3-s + 4·4-s + 6·5-s − 54·6-s − 40·7-s − 168·8-s + 81·9-s + 36·10-s − 564·11-s − 36·12-s − 638·13-s − 240·14-s − 54·15-s − 1.13e3·16-s + 882·17-s + 486·18-s + 24·20-s + 360·21-s − 3.38e3·22-s − 840·23-s + 1.51e3·24-s − 3.08e3·25-s − 3.82e3·26-s − 729·27-s − 160·28-s − 4.63e3·29-s + ⋯
L(s)  = 1  + 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.107·5-s − 0.612·6-s − 0.308·7-s − 0.928·8-s + 1/3·9-s + 0.113·10-s − 1.40·11-s − 0.0721·12-s − 1.04·13-s − 0.327·14-s − 0.0619·15-s − 1.10·16-s + 0.740·17-s + 0.353·18-s + 0.0134·20-s + 0.178·21-s − 1.49·22-s − 0.331·23-s + 0.535·24-s − 0.988·25-s − 1.11·26-s − 0.192·27-s − 0.0385·28-s − 1.02·29-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 0.80289878120.8028987812
L(12)L(\frac12) ≈\approx 0.80289878120.8028987812
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 1−3pT+p5T2 1 - 3 p T + p^{5} T^{2}
5 1−6T+p5T2 1 - 6 T + p^{5} T^{2}
7 1+40T+p5T2 1 + 40 T + p^{5} T^{2}
11 1+564T+p5T2 1 + 564 T + p^{5} T^{2}
13 1+638T+p5T2 1 + 638 T + p^{5} T^{2}
17 1−882T+p5T2 1 - 882 T + p^{5} T^{2}
23 1+840T+p5T2 1 + 840 T + p^{5} T^{2}
29 1+4638T+p5T2 1 + 4638 T + p^{5} T^{2}
31 1+4400T+p5T2 1 + 4400 T + p^{5} T^{2}
37 1−2410T+p5T2 1 - 2410 T + p^{5} T^{2}
41 1−6870T+p5T2 1 - 6870 T + p^{5} T^{2}
43 1−9644T+p5T2 1 - 9644 T + p^{5} T^{2}
47 1+18672T+p5T2 1 + 18672 T + p^{5} T^{2}
53 1+33750T+p5T2 1 + 33750 T + p^{5} T^{2}
59 1−18084T+p5T2 1 - 18084 T + p^{5} T^{2}
61 1−39758T+p5T2 1 - 39758 T + p^{5} T^{2}
67 1−23068T+p5T2 1 - 23068 T + p^{5} T^{2}
71 1−4248T+p5T2 1 - 4248 T + p^{5} T^{2}
73 1+41110T+p5T2 1 + 41110 T + p^{5} T^{2}
79 1+21920T+p5T2 1 + 21920 T + p^{5} T^{2}
83 1−82452T+p5T2 1 - 82452 T + p^{5} T^{2}
89 1−94086T+p5T2 1 - 94086 T + p^{5} T^{2}
97 1+49442T+p5T2 1 + 49442 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

−9.496836595909973790585042390430, −8.080345409244966479626722466801, −7.38917060485400188180780658399, −6.27266724183700216894881470675, −5.48241881589994775127136969577, −5.05460062120767708136580067217, −4.02609871485109633885335604064, −3.06423705626129388363328242660, −2.07827300670332858444113879625, −0.31498196263593192206757309694, 0.31498196263593192206757309694, 2.07827300670332858444113879625, 3.06423705626129388363328242660, 4.02609871485109633885335604064, 5.05460062120767708136580067217, 5.48241881589994775127136969577, 6.27266724183700216894881470675, 7.38917060485400188180780658399, 8.080345409244966479626722466801, 9.496836595909973790585042390430

Graph of the ZZ-function along the critical line