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Properties

Label 2-9576-1.1-c1-0-106
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  + 2·5-s − 7-s − 2·11-s − 3.12·13-s + 3.12·17-s − 19-s + 7.12·23-s − 25-s − 9.12·29-s − 1.12·31-s − 2·35-s − 0.876·37-s + 8.24·41-s + 4·43-s + 49-s − 5.12·53-s − 4·55-s − 6.24·59-s − 2·61-s − 6.24·65-s − 14.2·67-s + 9.36·71-s − 10·73-s + 2·77-s + 13.1·79-s + 9.12·83-s + 6.24·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 0.603·11-s − 0.866·13-s + 0.757·17-s − 0.229·19-s + 1.48·23-s − 0.200·25-s − 1.69·29-s − 0.201·31-s − 0.338·35-s − 0.144·37-s + 1.28·41-s + 0.609·43-s + 0.142·49-s − 0.703·53-s − 0.539·55-s − 0.813·59-s − 0.256·61-s − 0.774·65-s − 1.74·67-s + 1.11·71-s − 1.17·73-s + 0.227·77-s + 1.47·79-s + 1.00·83-s + 0.677·85-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 1−2T+5T2 1 - 2T + 5T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1+3.12T+13T2 1 + 3.12T + 13T^{2}
17 1−3.12T+17T2 1 - 3.12T + 17T^{2}
23 1−7.12T+23T2 1 - 7.12T + 23T^{2}
29 1+9.12T+29T2 1 + 9.12T + 29T^{2}
31 1+1.12T+31T2 1 + 1.12T + 31T^{2}
37 1+0.876T+37T2 1 + 0.876T + 37T^{2}
41 1−8.24T+41T2 1 - 8.24T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+5.12T+53T2 1 + 5.12T + 53T^{2}
59 1+6.24T+59T2 1 + 6.24T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+14.2T+67T2 1 + 14.2T + 67T^{2}
71 1−9.36T+71T2 1 - 9.36T + 71T^{2}
73 1+10T+73T2 1 + 10T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1−9.12T+83T2 1 - 9.12T + 83T^{2}
89 1−0.246T+89T2 1 - 0.246T + 89T^{2}
97 1−8.24T+97T2 1 - 8.24T + 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

−7.52976781513683755530328151294, −6.62165980050771849222973935862, −5.88257156651207040240560299792, −5.38710787028660319984339311543, −4.73928513783613514319005326355, −3.73972561851802654551471421950, −2.88819445948934469215030515990, −2.26631660693678829317883843248, −1.30198454774280321651165281559, 0, 1.30198454774280321651165281559, 2.26631660693678829317883843248, 2.88819445948934469215030515990, 3.73972561851802654551471421950, 4.73928513783613514319005326355, 5.38710787028660319984339311543, 5.88257156651207040240560299792, 6.62165980050771849222973935862, 7.52976781513683755530328151294

Graph of the ZZ-function along the critical line