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Properties

Label 2-2160-12.11-c1-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s − 1.26i·7-s − 1.26·11-s − 4.19·13-s − 5.19i·17-s + 6.46i·19-s + 7.73·23-s − 25-s + 2.19i·29-s + 6.46i·31-s + 1.26·35-s + 2·37-s + 8.19i·41-s + 11.6i·43-s − 2.53·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 0.479i·7-s − 0.382·11-s − 1.16·13-s − 1.26i·17-s + 1.48i·19-s + 1.61·23-s − 0.200·25-s + 0.407i·29-s + 1.16i·31-s + 0.214·35-s + 0.328·37-s + 1.28i·41-s + 1.77i·43-s − 0.369·47-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 1.1867577671.186757767
L(12)L(\frac12) ≈\approx 1.1867577671.186757767
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+1.26iT−7T2 1 + 1.26iT - 7T^{2}
11 1+1.26T+11T2 1 + 1.26T + 11T^{2}
13 1+4.19T+13T2 1 + 4.19T + 13T^{2}
17 1+5.19iT−17T2 1 + 5.19iT - 17T^{2}
19 1−6.46iT−19T2 1 - 6.46iT - 19T^{2}
23 1−7.73T+23T2 1 - 7.73T + 23T^{2}
29 1−2.19iT−29T2 1 - 2.19iT - 29T^{2}
31 1−6.46iT−31T2 1 - 6.46iT - 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1−8.19iT−41T2 1 - 8.19iT - 41T^{2}
43 1−11.6iT−43T2 1 - 11.6iT - 43T^{2}
47 1+2.53T+47T2 1 + 2.53T + 47T^{2}
53 1+0.803iT−53T2 1 + 0.803iT - 53T^{2}
59 1+11.6T+59T2 1 + 11.6T + 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1−10.3iT−67T2 1 - 10.3iT - 67T^{2}
71 1+1.26T+71T2 1 + 1.26T + 71T^{2}
73 1−6.19T+73T2 1 - 6.19T + 73T^{2}
79 1−3.92iT−79T2 1 - 3.92iT - 79T^{2}
83 1−5.19T+83T2 1 - 5.19T + 83T^{2}
89 1+3.80iT−89T2 1 + 3.80iT - 89T^{2}
97 1+8T+97T2 1 + 8T + 97T^{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

−9.478725381417883425772747196160, −8.413996668302737145806694325773, −7.51924885595354749275959020052, −7.13276125225537578475590837934, −6.21719957552703635874923542516, −5.13870464228729663608012801838, −4.58802398679959866710061836122, −3.28590817834723512813050154802, −2.67201338497142055084619081595, −1.23032000481983733540545183292, 0.43631070961811189632071913433, 2.03031818208954932468058188193, 2.83995562186468823654378980564, 4.07932918198831638818367437533, 4.97232893074457751734717892370, 5.53847089177282888754189162777, 6.58163375634899606061836509029, 7.37471491380295744333542600270, 8.119946811515155234991151682870, 9.080122044530265463173610861053

Graph of the ZZ-function along the critical line