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Properties

Label 2-6480-1.1-c1-0-23
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 − 1.43·7-s + 1.35·11-s − 5.52·13-s + 4.82·17-s + 0.648·19-s + 8.90·23-s + 25-s − 7.17·29-s + 4.64·31-s − 1.43·35-s + 1.35·37-s + 0.351·41-s − 4.82·43-s + 9.49·47-s − 4.93·49-s − 8.17·53-s + 1.35·55-s − 1.46·59-s + 6.69·61-s − 5.52·65-s − 12.4·67-s − 2.22·71-s − 4.34·73-s − 1.94·77-s + 13.0·79-s + 5.26·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.543·7-s + 0.407·11-s − 1.53·13-s + 1.16·17-s + 0.148·19-s + 1.85·23-s + 0.200·25-s − 1.33·29-s + 0.834·31-s − 0.243·35-s + 0.222·37-s + 0.0549·41-s − 0.735·43-s + 1.38·47-s − 0.704·49-s − 1.12·53-s + 0.182·55-s − 0.191·59-s + 0.857·61-s − 0.685·65-s − 1.51·67-s − 0.264·71-s − 0.508·73-s − 0.221·77-s + 1.46·79-s + 0.577·83-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 1.9520648311.952064831
L(12)L(\frac12) ≈\approx 1.9520648311.952064831
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+1.43T+7T2 1 + 1.43T + 7T^{2}
11 1−1.35T+11T2 1 - 1.35T + 11T^{2}
13 1+5.52T+13T2 1 + 5.52T + 13T^{2}
17 1−4.82T+17T2 1 - 4.82T + 17T^{2}
19 1−0.648T+19T2 1 - 0.648T + 19T^{2}
23 1−8.90T+23T2 1 - 8.90T + 23T^{2}
29 1+7.17T+29T2 1 + 7.17T + 29T^{2}
31 1−4.64T+31T2 1 - 4.64T + 31T^{2}
37 1−1.35T+37T2 1 - 1.35T + 37T^{2}
41 1−0.351T+41T2 1 - 0.351T + 41T^{2}
43 1+4.82T+43T2 1 + 4.82T + 43T^{2}
47 1−9.49T+47T2 1 - 9.49T + 47T^{2}
53 1+8.17T+53T2 1 + 8.17T + 53T^{2}
59 1+1.46T+59T2 1 + 1.46T + 59T^{2}
61 1−6.69T+61T2 1 - 6.69T + 61T^{2}
67 1+12.4T+67T2 1 + 12.4T + 67T^{2}
71 1+2.22T+71T2 1 + 2.22T + 71T^{2}
73 1+4.34T+73T2 1 + 4.34T + 73T^{2}
79 1−13.0T+79T2 1 - 13.0T + 79T^{2}
83 1−5.26T+83T2 1 - 5.26T + 83T^{2}
89 1+11T+89T2 1 + 11T + 89T^{2}
97 1−17.5T+97T2 1 - 17.5T + 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

−7.81014238881392944216608366839, −7.32417782246611439186240385764, −6.66304164824867870511030635526, −5.87153827298815125076737466634, −5.16355035873671785504375497858, −4.55974308653066974228749603232, −3.38913500103024881152654747512, −2.87252023059663371208334437680, −1.84573268714590542162019713095, −0.72408876703583060903983536965, 0.72408876703583060903983536965, 1.84573268714590542162019713095, 2.87252023059663371208334437680, 3.38913500103024881152654747512, 4.55974308653066974228749603232, 5.16355035873671785504375497858, 5.87153827298815125076737466634, 6.66304164824867870511030635526, 7.32417782246611439186240385764, 7.81014238881392944216608366839

Graph of the ZZ-function along the critical line