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Properties

Label 2-9576-1.1-c1-0-115
Degree 22
Conductor 95769576
Sign −1-1
Analytic cond. 76.464776.4647
Root an. cond. 8.744418.74441
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23·5-s + 7-s − 2.85·11-s − 3·13-s + 1.61·17-s − 19-s + 1.76·23-s + 0.854·29-s − 4.38·31-s + 2.23·35-s − 4.70·37-s + 1.14·41-s − 3.52·43-s + 7.47·47-s + 49-s − 10.3·53-s − 6.38·55-s + 3·59-s − 15.1·61-s − 6.70·65-s + 2.85·67-s + 15.1·71-s + 2.32·73-s − 2.85·77-s + 0.472·79-s − 9.56·83-s + 3.61·85-s + ⋯
L(s)  = 1  + 0.999·5-s + 0.377·7-s − 0.860·11-s − 0.832·13-s + 0.392·17-s − 0.229·19-s + 0.367·23-s + 0.158·29-s − 0.787·31-s + 0.377·35-s − 0.774·37-s + 0.178·41-s − 0.537·43-s + 1.08·47-s + 0.142·49-s − 1.41·53-s − 0.860·55-s + 0.390·59-s − 1.94·61-s − 0.832·65-s + 0.348·67-s + 1.80·71-s + 0.272·73-s − 0.325·77-s + 0.0531·79-s − 1.04·83-s + 0.392·85-s + ⋯

Functional equation

Λ(s)=(9576s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9576s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 95769576    =    23⋅32⋅7⋅192^{3} \cdot 3^{2} \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 76.464776.4647
Root analytic conductor: 8.744418.74441
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9576, ( :1/2), −1)(2,\ 9576,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
3 1 1
7 1−T 1 - T
19 1+T 1 + T
good5 1−2.23T+5T2 1 - 2.23T + 5T^{2}
11 1+2.85T+11T2 1 + 2.85T + 11T^{2}
13 1+3T+13T2 1 + 3T + 13T^{2}
17 1−1.61T+17T2 1 - 1.61T + 17T^{2}
23 1−1.76T+23T2 1 - 1.76T + 23T^{2}
29 1−0.854T+29T2 1 - 0.854T + 29T^{2}
31 1+4.38T+31T2 1 + 4.38T + 31T^{2}
37 1+4.70T+37T2 1 + 4.70T + 37T^{2}
41 1−1.14T+41T2 1 - 1.14T + 41T^{2}
43 1+3.52T+43T2 1 + 3.52T + 43T^{2}
47 1−7.47T+47T2 1 - 7.47T + 47T^{2}
53 1+10.3T+53T2 1 + 10.3T + 53T^{2}
59 1−3T+59T2 1 - 3T + 59T^{2}
61 1+15.1T+61T2 1 + 15.1T + 61T^{2}
67 1−2.85T+67T2 1 - 2.85T + 67T^{2}
71 1−15.1T+71T2 1 - 15.1T + 71T^{2}
73 1−2.32T+73T2 1 - 2.32T + 73T^{2}
79 1−0.472T+79T2 1 - 0.472T + 79T^{2}
83 1+9.56T+83T2 1 + 9.56T + 83T^{2}
89 1+5.70T+89T2 1 + 5.70T + 89T^{2}
97 1+9T+97T2 1 + 9T + 97T^{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

−7.43542493027224997737351180844, −6.63673183425818516865636857408, −5.89727521308227951996512601245, −5.23570394305607498854212253347, −4.85098158804577420470472341087, −3.80091515961461830913863109897, −2.83129301551225469370569455608, −2.19402882379646767640728040003, −1.39528374035394068376229403899, 0, 1.39528374035394068376229403899, 2.19402882379646767640728040003, 2.83129301551225469370569455608, 3.80091515961461830913863109897, 4.85098158804577420470472341087, 5.23570394305607498854212253347, 5.89727521308227951996512601245, 6.63673183425818516865636857408, 7.43542493027224997737351180844

Graph of the ZZ-function along the critical line