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Properties

Label 2-64800-1.1-c1-0-53
Degree 22
Conductor 6480064800
Sign −1-1
Analytic cond. 517.430517.430
Root an. cond. 22.747122.7471
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·7-s − 2·11-s + 2·13-s + 5·17-s − 2·19-s − 8·23-s − 10·37-s − 3·41-s + 2·43-s − 9·47-s + 2·49-s + 12·53-s − 6·59-s + 2·61-s + 12·67-s + 12·71-s − 13·73-s − 6·77-s − 79-s − 4·83-s + 7·89-s + 6·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.13·7-s − 0.603·11-s + 0.554·13-s + 1.21·17-s − 0.458·19-s − 1.66·23-s − 1.64·37-s − 0.468·41-s + 0.304·43-s − 1.31·47-s + 2/7·49-s + 1.64·53-s − 0.781·59-s + 0.256·61-s + 1.46·67-s + 1.42·71-s − 1.52·73-s − 0.683·77-s − 0.112·79-s − 0.439·83-s + 0.741·89-s + 0.628·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

Λ(s)=(64800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(64800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6480064800    =    25⋅34⋅522^{5} \cdot 3^{4} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 517.430517.430
Root analytic conductor: 22.747122.7471
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 64800, ( :1/2), −1)(2,\ 64800,\ (\ :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
5 1 1
good7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−5T+pT2 1 - 5 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+13T+pT2 1 + 13 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−7T+pT2 1 - 7 T + p T^{2}
97 1−T+pT2 1 - T + p T^{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

−14.35720906410047, −14.10270259674807, −13.56649312753488, −12.96284857279840, −12.43131381342055, −11.78366371023441, −11.66994266956723, −10.82159539912518, −10.48274649942577, −10.01036065297914, −9.448268602194141, −8.571946010162535, −8.292433552278649, −7.922272695374461, −7.304417210538045, −6.689892894125800, −5.949754136276323, −5.494452523000389, −5.008739455313877, −4.356646265266708, −3.701039579226852, −3.197435884620142, −2.184761312507507, −1.801751477178448, −1.013948729070745, 0, 1.013948729070745, 1.801751477178448, 2.184761312507507, 3.197435884620142, 3.701039579226852, 4.356646265266708, 5.008739455313877, 5.494452523000389, 5.949754136276323, 6.689892894125800, 7.304417210538045, 7.922272695374461, 8.292433552278649, 8.571946010162535, 9.448268602194141, 10.01036065297914, 10.48274649942577, 10.82159539912518, 11.66994266956723, 11.78366371023441, 12.43131381342055, 12.96284857279840, 13.56649312753488, 14.10270259674807, 14.35720906410047

Graph of the ZZ-function along the critical line