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Properties

Label 2-2160-1.1-c3-0-33
Degree 22
Conductor 21602160
Sign 11
Analytic cond. 127.444127.444
Root an. cond. 11.289111.2891
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 22·7-s + 12·11-s + 38·13-s − 105·17-s + 157·19-s + 117·23-s + 25·25-s + 66·29-s + 25·31-s − 110·35-s + 314·37-s − 504·41-s − 380·43-s + 252·47-s + 141·49-s + 3·53-s − 60·55-s + 318·59-s + 293·61-s − 190·65-s + 322·67-s + 120·71-s + 44·73-s + 264·77-s − 917·79-s − 309·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.18·7-s + 0.328·11-s + 0.810·13-s − 1.49·17-s + 1.89·19-s + 1.06·23-s + 1/5·25-s + 0.422·29-s + 0.144·31-s − 0.531·35-s + 1.39·37-s − 1.91·41-s − 1.34·43-s + 0.782·47-s + 0.411·49-s + 0.00777·53-s − 0.147·55-s + 0.701·59-s + 0.614·61-s − 0.362·65-s + 0.587·67-s + 0.200·71-s + 0.0705·73-s + 0.390·77-s − 1.30·79-s − 0.408·83-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 11
Analytic conductor: 127.444127.444
Root analytic conductor: 11.289111.2891
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2160, ( :3/2), 1)(2,\ 2160,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.7463515562.746351556
L(12)L(\frac12) ≈\approx 2.7463515562.746351556
L(52)L(\frac{5}{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+pT 1 + p T
good7 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
11 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
13 1−38T+p3T2 1 - 38 T + p^{3} T^{2}
17 1+105T+p3T2 1 + 105 T + p^{3} T^{2}
19 1−157T+p3T2 1 - 157 T + p^{3} T^{2}
23 1−117T+p3T2 1 - 117 T + p^{3} T^{2}
29 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
31 1−25T+p3T2 1 - 25 T + p^{3} T^{2}
37 1−314T+p3T2 1 - 314 T + p^{3} T^{2}
41 1+504T+p3T2 1 + 504 T + p^{3} T^{2}
43 1+380T+p3T2 1 + 380 T + p^{3} T^{2}
47 1−252T+p3T2 1 - 252 T + p^{3} T^{2}
53 1−3T+p3T2 1 - 3 T + p^{3} T^{2}
59 1−318T+p3T2 1 - 318 T + p^{3} T^{2}
61 1−293T+p3T2 1 - 293 T + p^{3} T^{2}
67 1−322T+p3T2 1 - 322 T + p^{3} T^{2}
71 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
73 1−44T+p3T2 1 - 44 T + p^{3} T^{2}
79 1+917T+p3T2 1 + 917 T + p^{3} T^{2}
83 1+309T+p3T2 1 + 309 T + p^{3} T^{2}
89 1−1272T+p3T2 1 - 1272 T + p^{3} T^{2}
97 1−1328T+p3T2 1 - 1328 T + p^{3} 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

−8.601591150513406660143105609660, −8.073984795241922239785229248594, −7.16796282077513354834222171152, −6.53246131072541104879264250647, −5.34185839777789793165124437247, −4.76298043839384812564481024238, −3.87223186896078592539082835511, −2.90333938493657302144748009078, −1.65853882783165728899243122584, −0.805261026022207019403575232192, 0.805261026022207019403575232192, 1.65853882783165728899243122584, 2.90333938493657302144748009078, 3.87223186896078592539082835511, 4.76298043839384812564481024238, 5.34185839777789793165124437247, 6.53246131072541104879264250647, 7.16796282077513354834222171152, 8.073984795241922239785229248594, 8.601591150513406660143105609660

Graph of the ZZ-function along the critical line