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Properties

Label 2-9576-1.1-c1-0-2
Degree 22
Conductor 95769576
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.23·5-s − 7-s + 0.763·11-s − 5.23·13-s − 2.76·17-s + 19-s − 1.23·23-s + 5.47·25-s + 0.472·29-s − 8.47·31-s + 3.23·35-s − 8.94·37-s − 2·41-s − 4·43-s − 2.47·47-s + 49-s − 8.47·53-s − 2.47·55-s + 8·59-s − 0.472·61-s + 16.9·65-s + 5.23·67-s + 10.4·71-s − 4.47·73-s − 0.763·77-s + 2.29·79-s − 2·83-s + ⋯
L(s)  = 1  − 1.44·5-s − 0.377·7-s + 0.230·11-s − 1.45·13-s − 0.670·17-s + 0.229·19-s − 0.257·23-s + 1.09·25-s + 0.0876·29-s − 1.52·31-s + 0.546·35-s − 1.47·37-s − 0.312·41-s − 0.609·43-s − 0.360·47-s + 0.142·49-s − 1.16·53-s − 0.333·55-s + 1.04·59-s − 0.0604·61-s + 2.10·65-s + 0.639·67-s + 1.24·71-s − 0.523·73-s − 0.0870·77-s + 0.257·79-s − 0.219·83-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: 11
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: 00
Selberg data: (2, 9576, ( :1/2), 1)(2,\ 9576,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.34588665540.3458866554
L(12)L(\frac12) ≈\approx 0.34588665540.3458866554
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+3.23T+5T2 1 + 3.23T + 5T^{2}
11 1−0.763T+11T2 1 - 0.763T + 11T^{2}
13 1+5.23T+13T2 1 + 5.23T + 13T^{2}
17 1+2.76T+17T2 1 + 2.76T + 17T^{2}
23 1+1.23T+23T2 1 + 1.23T + 23T^{2}
29 1−0.472T+29T2 1 - 0.472T + 29T^{2}
31 1+8.47T+31T2 1 + 8.47T + 31T^{2}
37 1+8.94T+37T2 1 + 8.94T + 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+2.47T+47T2 1 + 2.47T + 47T^{2}
53 1+8.47T+53T2 1 + 8.47T + 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1+0.472T+61T2 1 + 0.472T + 61T^{2}
67 1−5.23T+67T2 1 - 5.23T + 67T^{2}
71 1−10.4T+71T2 1 - 10.4T + 71T^{2}
73 1+4.47T+73T2 1 + 4.47T + 73T^{2}
79 1−2.29T+79T2 1 - 2.29T + 79T^{2}
83 1+2T+83T2 1 + 2T + 83T^{2}
89 1+6.94T+89T2 1 + 6.94T + 89T^{2}
97 1+3.23T+97T2 1 + 3.23T + 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

−7.69247814917445102402354737473, −6.95761701017399195808708798476, −6.69962296376422786067278444002, −5.44107036732067245304075133221, −4.91549410761598101225538044907, −4.07253397121681352589706421983, −3.56455118970851291484025486510, −2.73826941560852841838158480692, −1.74955773910235877915909269219, −0.26877759296438580179967345213, 0.26877759296438580179967345213, 1.74955773910235877915909269219, 2.73826941560852841838158480692, 3.56455118970851291484025486510, 4.07253397121681352589706421983, 4.91549410761598101225538044907, 5.44107036732067245304075133221, 6.69962296376422786067278444002, 6.95761701017399195808708798476, 7.69247814917445102402354737473

Graph of the ZZ-function along the critical line