↑

Properties

Label 2-2160-12.11-c1-0-9
Degree 22
Conductor 21602160
Sign −i-i
Analytic cond. 17.247617.2476
Root an. cond. 4.153034.15303
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·5-s + 4.73i·7-s − 4.73·11-s + 6.19·13-s − 5.19i·17-s + 0.464i·19-s + 4.26·23-s − 25-s + 8.19i·29-s + 0.464i·31-s + 4.73·35-s + 2·37-s + 2.19i·41-s + 5.66i·43-s − 9.46·47-s + ⋯
L(s)  = 1  − 0.447i·5-s + 1.78i·7-s − 1.42·11-s + 1.71·13-s − 1.26i·17-s + 0.106i·19-s + 0.889·23-s − 0.200·25-s + 1.52i·29-s + 0.0833i·31-s + 0.799·35-s + 0.328·37-s + 0.342i·41-s + 0.863i·43-s − 1.38·47-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −i-i
Analytic conductor: 17.247617.2476
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2160(431,⋅)\chi_{2160} (431, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1/2), −i)(2,\ 2160,\ (\ :1/2),\ -i)

Particular Values

L(1)L(1) ≈\approx 1.4209069871.420906987
L(12)L(\frac12) ≈\approx 1.4209069871.420906987
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+iT 1 + iT
good7 1−4.73iT−7T2 1 - 4.73iT - 7T^{2}
11 1+4.73T+11T2 1 + 4.73T + 11T^{2}
13 1−6.19T+13T2 1 - 6.19T + 13T^{2}
17 1+5.19iT−17T2 1 + 5.19iT - 17T^{2}
19 1−0.464iT−19T2 1 - 0.464iT - 19T^{2}
23 1−4.26T+23T2 1 - 4.26T + 23T^{2}
29 1−8.19iT−29T2 1 - 8.19iT - 29T^{2}
31 1−0.464iT−31T2 1 - 0.464iT - 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1−2.19iT−41T2 1 - 2.19iT - 41T^{2}
43 1−5.66iT−43T2 1 - 5.66iT - 43T^{2}
47 1+9.46T+47T2 1 + 9.46T + 47T^{2}
53 1−11.1iT−53T2 1 - 11.1iT - 53T^{2}
59 1−5.66T+59T2 1 - 5.66T + 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1−10.3iT−67T2 1 - 10.3iT - 67T^{2}
71 1+4.73T+71T2 1 + 4.73T + 71T^{2}
73 1+4.19T+73T2 1 + 4.19T + 73T^{2}
79 1−9.92iT−79T2 1 - 9.92iT - 79T^{2}
83 1+5.19T+83T2 1 + 5.19T + 83T^{2}
89 1−14.1iT−89T2 1 - 14.1iT - 89T^{2}
97 1+8T+97T2 1 + 8T + 97T^{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

−9.052813649819912957001489825625, −8.605120203779839910341464199134, −7.968977223843871554604815401697, −6.88751746539785423955840514277, −5.87922755984119348496344402269, −5.37626723346144320415770465442, −4.69909309616188821483787809321, −3.18986345766012403224163924120, −2.62688782480522439807356017849, −1.31398808466998804631050492899, 0.52576368104977398623015708558, 1.80484765349897089130126594125, 3.24554581397598036876368584785, 3.84678764616181897334961248691, 4.72271820736228343799453897253, 5.88089786954278546467640145991, 6.54801771529231945686448002656, 7.41428225271384938875282564884, 8.019484424208002091569181280772, 8.672676622884276120883968149444

Graph of the ZZ-function along the critical line