↑

Properties

Label 2-2160-1.1-c3-0-28
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 16.1·7-s + 55.5·11-s + 43.4·13-s + 25.5·17-s + 103.·19-s + 4.47·23-s + 25·25-s − 4.47·29-s + 45.1·31-s + 80.5·35-s + 69.5·37-s − 483.·41-s − 151.·43-s − 67.6·47-s − 83.6·49-s − 278.·53-s − 277.·55-s + 257.·59-s − 489.·61-s − 217.·65-s + 772.·67-s + 536.·71-s − 65.5·73-s − 894.·77-s − 749.·79-s + 1.26e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.869·7-s + 1.52·11-s + 0.926·13-s + 0.364·17-s + 1.24·19-s + 0.0405·23-s + 0.200·25-s − 0.0286·29-s + 0.261·31-s + 0.388·35-s + 0.309·37-s − 1.84·41-s − 0.538·43-s − 0.209·47-s − 0.243·49-s − 0.722·53-s − 0.680·55-s + 0.569·59-s − 1.02·61-s − 0.414·65-s + 1.40·67-s + 0.897·71-s − 0.105·73-s − 1.32·77-s − 1.06·79-s + 1.66·83-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 2.1315383812.131538381
L(12)L(\frac12) ≈\approx 2.1315383812.131538381
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+16.1T+343T2 1 + 16.1T + 343T^{2}
11 1−55.5T+1.33e3T2 1 - 55.5T + 1.33e3T^{2}
13 1−43.4T+2.19e3T2 1 - 43.4T + 2.19e3T^{2}
17 1−25.5T+4.91e3T2 1 - 25.5T + 4.91e3T^{2}
19 1−103.T+6.85e3T2 1 - 103.T + 6.85e3T^{2}
23 1−4.47T+1.21e4T2 1 - 4.47T + 1.21e4T^{2}
29 1+4.47T+2.43e4T2 1 + 4.47T + 2.43e4T^{2}
31 1−45.1T+2.97e4T2 1 - 45.1T + 2.97e4T^{2}
37 1−69.5T+5.06e4T2 1 - 69.5T + 5.06e4T^{2}
41 1+483.T+6.89e4T2 1 + 483.T + 6.89e4T^{2}
43 1+151.T+7.95e4T2 1 + 151.T + 7.95e4T^{2}
47 1+67.6T+1.03e5T2 1 + 67.6T + 1.03e5T^{2}
53 1+278.T+1.48e5T2 1 + 278.T + 1.48e5T^{2}
59 1−257.T+2.05e5T2 1 - 257.T + 2.05e5T^{2}
61 1+489.T+2.26e5T2 1 + 489.T + 2.26e5T^{2}
67 1−772.T+3.00e5T2 1 - 772.T + 3.00e5T^{2}
71 1−536.T+3.57e5T2 1 - 536.T + 3.57e5T^{2}
73 1+65.5T+3.89e5T2 1 + 65.5T + 3.89e5T^{2}
79 1+749.T+4.93e5T2 1 + 749.T + 4.93e5T^{2}
83 1−1.26e3T+5.71e5T2 1 - 1.26e3T + 5.71e5T^{2}
89 1+1.32e3T+7.04e5T2 1 + 1.32e3T + 7.04e5T^{2}
97 1+32.9T+9.12e5T2 1 + 32.9T + 9.12e5T^{2}
show more
show less
   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

−8.743050330787032800364311651554, −8.024798881901938031021181887621, −6.97868333190490676362477793761, −6.51319642660751826677872206699, −5.67664306302585915424190799861, −4.60019492804469337295226312184, −3.53557814764942724806714926623, −3.27512983482022174398700754904, −1.62317865142806589866639950762, −0.69993005163929511220152804932, 0.69993005163929511220152804932, 1.62317865142806589866639950762, 3.27512983482022174398700754904, 3.53557814764942724806714926623, 4.60019492804469337295226312184, 5.67664306302585915424190799861, 6.51319642660751826677872206699, 6.97868333190490676362477793761, 8.024798881901938031021181887621, 8.743050330787032800364311651554

Graph of the ZZ-function along the critical line