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Properties

Label 2-2160-1.1-c3-0-24
Degree 22
Conductor 21602160
Sign 11
Analytic cond. 127.444127.444
Root an. cond. 11.289111.2891
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 10.2·7-s + 44.3·11-s − 50.8·13-s + 25.2·17-s + 31.3·19-s − 76.1·23-s + 25·25-s + 156.·29-s − 134.·31-s − 51.2·35-s + 81.2·37-s + 326.·41-s − 422.·43-s + 452.·47-s − 237.·49-s + 98.1·53-s + 221.·55-s − 540.·59-s + 522.·61-s − 254.·65-s + 129.·67-s − 26.6·71-s − 147.·73-s − 454.·77-s + 1.08e3·79-s − 594.·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.553·7-s + 1.21·11-s − 1.08·13-s + 0.360·17-s + 0.378·19-s − 0.690·23-s + 0.200·25-s + 0.998·29-s − 0.779·31-s − 0.247·35-s + 0.360·37-s + 1.24·41-s − 1.49·43-s + 1.40·47-s − 0.693·49-s + 0.254·53-s + 0.543·55-s − 1.19·59-s + 1.09·61-s − 0.485·65-s + 0.235·67-s − 0.0444·71-s − 0.237·73-s − 0.673·77-s + 1.55·79-s − 0.786·83-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: no
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 2.2181443262.218144326
L(12)L(\frac12) ≈\approx 2.2181443262.218144326
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 1−5T 1 - 5T
good7 1+10.2T+343T2 1 + 10.2T + 343T^{2}
11 1−44.3T+1.33e3T2 1 - 44.3T + 1.33e3T^{2}
13 1+50.8T+2.19e3T2 1 + 50.8T + 2.19e3T^{2}
17 1−25.2T+4.91e3T2 1 - 25.2T + 4.91e3T^{2}
19 1−31.3T+6.85e3T2 1 - 31.3T + 6.85e3T^{2}
23 1+76.1T+1.21e4T2 1 + 76.1T + 1.21e4T^{2}
29 1−156.T+2.43e4T2 1 - 156.T + 2.43e4T^{2}
31 1+134.T+2.97e4T2 1 + 134.T + 2.97e4T^{2}
37 1−81.2T+5.06e4T2 1 - 81.2T + 5.06e4T^{2}
41 1−326.T+6.89e4T2 1 - 326.T + 6.89e4T^{2}
43 1+422.T+7.95e4T2 1 + 422.T + 7.95e4T^{2}
47 1−452.T+1.03e5T2 1 - 452.T + 1.03e5T^{2}
53 1−98.1T+1.48e5T2 1 - 98.1T + 1.48e5T^{2}
59 1+540.T+2.05e5T2 1 + 540.T + 2.05e5T^{2}
61 1−522.T+2.26e5T2 1 - 522.T + 2.26e5T^{2}
67 1−129.T+3.00e5T2 1 - 129.T + 3.00e5T^{2}
71 1+26.6T+3.57e5T2 1 + 26.6T + 3.57e5T^{2}
73 1+147.T+3.89e5T2 1 + 147.T + 3.89e5T^{2}
79 1−1.08e3T+4.93e5T2 1 - 1.08e3T + 4.93e5T^{2}
83 1+594.T+5.71e5T2 1 + 594.T + 5.71e5T^{2}
89 1−592.T+7.04e5T2 1 - 592.T + 7.04e5T^{2}
97 1+666.T+9.12e5T2 1 + 666.T + 9.12e5T^{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

−8.893264620884909463560142931877, −7.900709345842155104201924872688, −7.06946208982433766719888245251, −6.40353320112490616042641398705, −5.64643679931311375098650570035, −4.69883424859375951691219361431, −3.78712464588099908583269822776, −2.84300705714419785474799367710, −1.82454083217973335123660263679, −0.67727262051236517908123904869, 0.67727262051236517908123904869, 1.82454083217973335123660263679, 2.84300705714419785474799367710, 3.78712464588099908583269822776, 4.69883424859375951691219361431, 5.64643679931311375098650570035, 6.40353320112490616042641398705, 7.06946208982433766719888245251, 7.900709345842155104201924872688, 8.893264620884909463560142931877

Graph of the ZZ-function along the critical line