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Properties

Label 2-2160-1.1-c3-0-19
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 + 6·7-s − 47·11-s − 5·13-s + 131·17-s + 56·19-s + 3·23-s + 25·25-s + 157·29-s − 225·31-s − 30·35-s − 70·37-s − 140·41-s − 397·43-s − 347·47-s − 307·49-s − 4·53-s + 235·55-s + 748·59-s − 338·61-s + 25·65-s − 492·67-s + 32·71-s + 970·73-s − 282·77-s + 1.25e3·79-s − 102·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.323·7-s − 1.28·11-s − 0.106·13-s + 1.86·17-s + 0.676·19-s + 0.0271·23-s + 1/5·25-s + 1.00·29-s − 1.30·31-s − 0.144·35-s − 0.311·37-s − 0.533·41-s − 1.40·43-s − 1.07·47-s − 0.895·49-s − 0.0103·53-s + 0.576·55-s + 1.65·59-s − 0.709·61-s + 0.0477·65-s − 0.897·67-s + 0.0534·71-s + 1.55·73-s − 0.417·77-s + 1.79·79-s − 0.134·83-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 1.7367516501.736751650
L(12)L(\frac12) ≈\approx 1.7367516501.736751650
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 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1+47T+p3T2 1 + 47 T + p^{3} T^{2}
13 1+5T+p3T2 1 + 5 T + p^{3} T^{2}
17 1−131T+p3T2 1 - 131 T + p^{3} T^{2}
19 1−56T+p3T2 1 - 56 T + p^{3} T^{2}
23 1−3T+p3T2 1 - 3 T + p^{3} T^{2}
29 1−157T+p3T2 1 - 157 T + p^{3} T^{2}
31 1+225T+p3T2 1 + 225 T + p^{3} T^{2}
37 1+70T+p3T2 1 + 70 T + p^{3} T^{2}
41 1+140T+p3T2 1 + 140 T + p^{3} T^{2}
43 1+397T+p3T2 1 + 397 T + p^{3} T^{2}
47 1+347T+p3T2 1 + 347 T + p^{3} T^{2}
53 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
59 1−748T+p3T2 1 - 748 T + p^{3} T^{2}
61 1+338T+p3T2 1 + 338 T + p^{3} T^{2}
67 1+492T+p3T2 1 + 492 T + p^{3} T^{2}
71 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
73 1−970T+p3T2 1 - 970 T + p^{3} T^{2}
79 1−1257T+p3T2 1 - 1257 T + p^{3} T^{2}
83 1+102T+p3T2 1 + 102 T + p^{3} T^{2}
89 1−1488T+p3T2 1 - 1488 T + p^{3} T^{2}
97 1−974T+p3T2 1 - 974 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

−8.498100785907236993859845443436, −7.890196878816479062806280722872, −7.42669317580197836844514959844, −6.39599039214083899139911304265, −5.16037531041611192514165527988, −5.10589647059892854649800396385, −3.60775903425659351766426725215, −3.03166797054684620898646913165, −1.77048020943707349065105844139, −0.59887782828199126544780922026, 0.59887782828199126544780922026, 1.77048020943707349065105844139, 3.03166797054684620898646913165, 3.60775903425659351766426725215, 5.10589647059892854649800396385, 5.16037531041611192514165527988, 6.39599039214083899139911304265, 7.42669317580197836844514959844, 7.890196878816479062806280722872, 8.498100785907236993859845443436

Graph of the ZZ-function along the critical line