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Properties

Label 2-5760-1.1-c1-0-5
Degree 22
Conductor 57605760
Sign 11
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 6·11-s + 2·13-s + 2·17-s + 2·19-s − 6·23-s + 25-s + 10·29-s − 4·31-s + 2·35-s − 2·37-s − 6·41-s − 6·47-s − 3·49-s − 10·53-s + 6·55-s + 6·59-s − 6·61-s − 2·65-s + 16·67-s − 8·71-s + 10·73-s + 12·77-s − 12·83-s − 2·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 1.80·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 1.85·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s − 0.937·41-s − 0.875·47-s − 3/7·49-s − 1.37·53-s + 0.809·55-s + 0.781·59-s − 0.768·61-s − 0.248·65-s + 1.95·67-s − 0.949·71-s + 1.17·73-s + 1.36·77-s − 1.31·83-s − 0.216·85-s − 0.635·89-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), 1)(2,\ 5760,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.98280260960.9828026096
L(12)L(\frac12) ≈\approx 0.98280260960.9828026096
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+2T+pT2 1 + 2 T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−8.176599100732698173396854839843, −7.50291136652528562810030125950, −6.69830524006425872751699552453, −5.97915205263870641362947890885, −5.23460451980427137977880559325, −4.53745308849339692219713158316, −3.41526557487911167785288991100, −3.03645100630012556525198927110, −1.93295809815488958901022988112, −0.50378114550564862969297887817, 0.50378114550564862969297887817, 1.93295809815488958901022988112, 3.03645100630012556525198927110, 3.41526557487911167785288991100, 4.53745308849339692219713158316, 5.23460451980427137977880559325, 5.97915205263870641362947890885, 6.69830524006425872751699552453, 7.50291136652528562810030125950, 8.176599100732698173396854839843

Graph of the ZZ-function along the critical line