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Properties

Label 2-1152-8.3-c4-0-49
Degree 22
Conductor 11521152
Sign 11
Analytic cond. 119.082119.082
Root an. cond. 10.912410.9124
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 30.1i·5-s + 52.3i·7-s − 90.0·11-s − 60.3i·13-s + 338·17-s + 6.92·19-s − 732. i·23-s − 287·25-s − 1.29e3i·29-s − 1.30e3i·31-s − 1.57e3·35-s − 241. i·37-s − 578·41-s + 2.02e3·43-s + 2.19e3i·47-s + ⋯
L(s)  = 1  + 1.20i·5-s + 1.06i·7-s − 0.744·11-s − 0.357i·13-s + 1.16·17-s + 0.0191·19-s − 1.38i·23-s − 0.459·25-s − 1.54i·29-s − 1.36i·31-s − 1.28·35-s − 0.176i·37-s − 0.343·41-s + 1.09·43-s + 0.994i·47-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=( Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+2) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 11
Analytic conductor: 119.082119.082
Root analytic conductor: 10.912410.9124
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ1152(703,⋅)\chi_{1152} (703, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :2), 1)(2,\ 1152,\ (\ :2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.8829663721.882966372
L(12)L(\frac12) ≈\approx 1.8829663721.882966372
L(3)L(3) 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
good5 1−30.1iT−625T2 1 - 30.1iT - 625T^{2}
7 1−52.3iT−2.40e3T2 1 - 52.3iT - 2.40e3T^{2}
11 1+90.0T+1.46e4T2 1 + 90.0T + 1.46e4T^{2}
13 1+60.3iT−2.85e4T2 1 + 60.3iT - 2.85e4T^{2}
17 1−338T+8.35e4T2 1 - 338T + 8.35e4T^{2}
19 1−6.92T+1.30e5T2 1 - 6.92T + 1.30e5T^{2}
23 1+732.iT−2.79e5T2 1 + 732. iT - 2.79e5T^{2}
29 1+1.29e3iT−7.07e5T2 1 + 1.29e3iT - 7.07e5T^{2}
31 1+1.30e3iT−9.23e5T2 1 + 1.30e3iT - 9.23e5T^{2}
37 1+241.iT−1.87e6T2 1 + 241. iT - 1.87e6T^{2}
41 1+578T+2.82e6T2 1 + 578T + 2.82e6T^{2}
43 1−2.02e3T+3.41e6T2 1 - 2.02e3T + 3.41e6T^{2}
47 1−2.19e3iT−4.87e6T2 1 - 2.19e3iT - 4.87e6T^{2}
53 1+2.44e3iT−7.89e6T2 1 + 2.44e3iT - 7.89e6T^{2}
59 1+1.19e3T+1.21e7T2 1 + 1.19e3T + 1.21e7T^{2}
61 1+6.40e3iT−1.38e7T2 1 + 6.40e3iT - 1.38e7T^{2}
67 1−8.26e3T+2.01e7T2 1 - 8.26e3T + 2.01e7T^{2}
71 1+4.28e3iT−2.54e7T2 1 + 4.28e3iT - 2.54e7T^{2}
73 1−8.73e3T+2.83e7T2 1 - 8.73e3T + 2.83e7T^{2}
79 1−1.12e4iT−3.89e7T2 1 - 1.12e4iT - 3.89e7T^{2}
83 1+1.31e4T+4.74e7T2 1 + 1.31e4T + 4.74e7T^{2}
89 1+910T+6.27e7T2 1 + 910T + 6.27e7T^{2}
97 1−5.42e3T+8.85e7T2 1 - 5.42e3T + 8.85e7T^{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

−9.424332799863417280140971221189, −8.164479743490122971276960061213, −7.78804796839110621378566200477, −6.61674474589300111335588745678, −5.94686567808535132700306766262, −5.15311585862818439236687723050, −3.84302301657577826974050465810, −2.71984505465308676734964285526, −2.32645304898405767077708900964, −0.49336405602045809382064856743, 0.862731231821676076865272957900, 1.51265736608282480410452283864, 3.15158485786315712760027768100, 4.06327483745814172307957654558, 5.06481134329232952090541124165, 5.55800104919344542821395737093, 6.98887869751738839258999824203, 7.57277666712569761963586487521, 8.462973402936904680033032568112, 9.183753411862665859634823558426

Graph of the ZZ-function along the critical line