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Properties

Label 2-6480-1.1-c1-0-22
Degree 22
Conductor 64806480
Sign 11
Analytic cond. 51.743051.7430
Root an. cond. 7.193267.19326
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  + 5-s − 4.27·7-s + 1.27·11-s + 6.27·13-s + 2·17-s − 19-s − 0.274·23-s + 25-s − 1.27·29-s + 1.27·31-s − 4.27·35-s + 4.54·37-s − 7.54·41-s − 4·43-s − 6.27·47-s + 11.2·49-s + 8.27·53-s + 1.27·55-s + 13·59-s − 6.54·61-s + 6.27·65-s + 14.5·67-s + 0.725·71-s − 15.0·73-s − 5.45·77-s + 4.54·79-s − 12.5·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.61·7-s + 0.384·11-s + 1.74·13-s + 0.485·17-s − 0.229·19-s − 0.0573·23-s + 0.200·25-s − 0.236·29-s + 0.228·31-s − 0.722·35-s + 0.747·37-s − 1.17·41-s − 0.609·43-s − 0.915·47-s + 1.61·49-s + 1.13·53-s + 0.171·55-s + 1.69·59-s − 0.838·61-s + 0.778·65-s + 1.77·67-s + 0.0860·71-s − 1.76·73-s − 0.621·77-s + 0.511·79-s − 1.37·83-s + ⋯

Functional equation

Λ(s)=(6480s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6480s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 64806480    =    24⋅34⋅52^{4} \cdot 3^{4} \cdot 5
Sign: 11
Analytic conductor: 51.743051.7430
Root analytic conductor: 7.193267.19326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6480, ( :1/2), 1)(2,\ 6480,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8945913121.894591312
L(12)L(\frac12) ≈\approx 1.8945913121.894591312
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
5 1−T 1 - T
good7 1+4.27T+7T2 1 + 4.27T + 7T^{2}
11 1−1.27T+11T2 1 - 1.27T + 11T^{2}
13 1−6.27T+13T2 1 - 6.27T + 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+T+19T2 1 + T + 19T^{2}
23 1+0.274T+23T2 1 + 0.274T + 23T^{2}
29 1+1.27T+29T2 1 + 1.27T + 29T^{2}
31 1−1.27T+31T2 1 - 1.27T + 31T^{2}
37 1−4.54T+37T2 1 - 4.54T + 37T^{2}
41 1+7.54T+41T2 1 + 7.54T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+6.27T+47T2 1 + 6.27T + 47T^{2}
53 1−8.27T+53T2 1 - 8.27T + 53T^{2}
59 1−13T+59T2 1 - 13T + 59T^{2}
61 1+6.54T+61T2 1 + 6.54T + 61T^{2}
67 1−14.5T+67T2 1 - 14.5T + 67T^{2}
71 1−0.725T+71T2 1 - 0.725T + 71T^{2}
73 1+15.0T+73T2 1 + 15.0T + 73T^{2}
79 1−4.54T+79T2 1 - 4.54T + 79T^{2}
83 1+12.5T+83T2 1 + 12.5T + 83T^{2}
89 1+9.82T+89T2 1 + 9.82T + 89T^{2}
97 1−16T+97T2 1 - 16T + 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

−8.184983694435237956185362496476, −7.04128196177843053936631471924, −6.54768077981291665050237455404, −6.01106787266463468182907304697, −5.41308993567024169202671242386, −4.18487038051319654468145275660, −3.52527771959397135454840513649, −2.95712391446607568717122170974, −1.77463562754797208096955677276, −0.72300981464788957430510522089, 0.72300981464788957430510522089, 1.77463562754797208096955677276, 2.95712391446607568717122170974, 3.52527771959397135454840513649, 4.18487038051319654468145275660, 5.41308993567024169202671242386, 6.01106787266463468182907304697, 6.54768077981291665050237455404, 7.04128196177843053936631471924, 8.184983694435237956185362496476

Graph of the ZZ-function along the critical line