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Properties

Label 2-76e2-1.1-c1-0-48
Degree 22
Conductor 57765776
Sign 11
Analytic cond. 46.121546.1215
Root an. cond. 6.791286.79128
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·9-s + 4·11-s − 13-s − 15-s + 3·17-s − 5·23-s − 4·25-s − 5·27-s + 7·29-s − 4·31-s + 4·33-s + 10·37-s − 39-s − 5·41-s + 5·43-s + 2·45-s + 7·47-s − 7·49-s + 3·51-s + 11·53-s − 4·55-s − 3·59-s + 11·61-s + 65-s + 3·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.727·17-s − 1.04·23-s − 4/5·25-s − 0.962·27-s + 1.29·29-s − 0.718·31-s + 0.696·33-s + 1.64·37-s − 0.160·39-s − 0.780·41-s + 0.762·43-s + 0.298·45-s + 1.02·47-s − 49-s + 0.420·51-s + 1.51·53-s − 0.539·55-s − 0.390·59-s + 1.40·61-s + 0.124·65-s + 0.366·67-s + ⋯

Functional equation

Λ(s)=(5776s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5776s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57765776    =    24⋅1922^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 46.121546.1215
Root analytic conductor: 6.791286.79128
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5776, ( :1/2), 1)(2,\ 5776,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1012103922.101210392
L(12)L(\frac12) ≈\approx 2.1012103922.101210392
L(32)L(\frac{3}{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)
bad2 1 1
19 1 1
good3 1−T+pT2 1 - T + p T^{2}
5 1+T+pT2 1 + T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
23 1+5T+pT2 1 + 5 T + p T^{2}
29 1−7T+pT2 1 - 7 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1+5T+pT2 1 + 5 T + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1−7T+pT2 1 - 7 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1−11T+pT2 1 - 11 T + p T^{2}
67 1−3T+pT2 1 - 3 T + p T^{2}
71 1+11T+pT2 1 + 11 T + p T^{2}
73 1−15T+pT2 1 - 15 T + p T^{2}
79 1−13T+pT2 1 - 13 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−3T+pT2 1 - 3 T + p T^{2}
97 1+5T+pT2 1 + 5 T + p 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

−8.070223875508849209271112014483, −7.63607991647119737095362961926, −6.69614388410678726347088668218, −6.02249493803915114497163101839, −5.28968462197699658502621172591, −4.14806492488521143828856305471, −3.76507390629787346769837834466, −2.83397001973401195812410569621, −1.98100308025709555986426797089, −0.74353520954790544832755635907, 0.74353520954790544832755635907, 1.98100308025709555986426797089, 2.83397001973401195812410569621, 3.76507390629787346769837834466, 4.14806492488521143828856305471, 5.28968462197699658502621172591, 6.02249493803915114497163101839, 6.69614388410678726347088668218, 7.63607991647119737095362961926, 8.070223875508849209271112014483

Graph of the ZZ-function along the critical line