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Properties

Label 2-1152-8.5-c3-0-6
Degree 22
Conductor 11521152
Sign −i-i
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.3i·5-s − 14.6·7-s + 5.65i·11-s + 17·25-s − 218. i·29-s − 338.·31-s + 152. i·35-s − 127·49-s + 509. i·53-s + 58.7·55-s + 554. i·59-s + 322·73-s − 83.1i·77-s + 308.·79-s + 1.22e3i·83-s + ⋯
L(s)  = 1  − 0.929i·5-s − 0.793·7-s + 0.155i·11-s + 0.136·25-s − 1.39i·29-s − 1.95·31-s + 0.737i·35-s − 0.370·49-s + 1.31i·53-s + 0.144·55-s + 1.22i·59-s + 0.516·73-s − 0.123i·77-s + 0.439·79-s + 1.62i·83-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=(−i Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=(−i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: −i-i
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1152(577,⋅)\chi_{1152} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), −i)(2,\ 1152,\ (\ :3/2),\ -i)

Particular Values

L(2)L(2) ≈\approx 0.68643877520.6864387752
L(12)L(\frac12) ≈\approx 0.68643877520.6864387752
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
good5 1+10.3iT−125T2 1 + 10.3iT - 125T^{2}
7 1+14.6T+343T2 1 + 14.6T + 343T^{2}
11 1−5.65iT−1.33e3T2 1 - 5.65iT - 1.33e3T^{2}
13 1−2.19e3T2 1 - 2.19e3T^{2}
17 1+4.91e3T2 1 + 4.91e3T^{2}
19 1−6.85e3T2 1 - 6.85e3T^{2}
23 1+1.21e4T2 1 + 1.21e4T^{2}
29 1+218.iT−2.43e4T2 1 + 218. iT - 2.43e4T^{2}
31 1+338.T+2.97e4T2 1 + 338.T + 2.97e4T^{2}
37 1−5.06e4T2 1 - 5.06e4T^{2}
41 1+6.89e4T2 1 + 6.89e4T^{2}
43 1−7.95e4T2 1 - 7.95e4T^{2}
47 1+1.03e5T2 1 + 1.03e5T^{2}
53 1−509.iT−1.48e5T2 1 - 509. iT - 1.48e5T^{2}
59 1−554.iT−2.05e5T2 1 - 554. iT - 2.05e5T^{2}
61 1−2.26e5T2 1 - 2.26e5T^{2}
67 1−3.00e5T2 1 - 3.00e5T^{2}
71 1+3.57e5T2 1 + 3.57e5T^{2}
73 1−322T+3.89e5T2 1 - 322T + 3.89e5T^{2}
79 1−308.T+4.93e5T2 1 - 308.T + 4.93e5T^{2}
83 1−1.22e3iT−5.71e5T2 1 - 1.22e3iT - 5.71e5T^{2}
89 1+7.04e5T2 1 + 7.04e5T^{2}
97 1+574T+9.12e5T2 1 + 574T + 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

−9.441693097634920580878082873735, −8.994071311433229844906527274484, −8.024461881209783311379158359721, −7.18188474678392013337117004315, −6.19473252602108113973153109669, −5.39774657732659590211850382896, −4.43200511806724335986078499005, −3.52243760349827180664154857187, −2.28398610328640160690189559074, −0.982462127589477152619674389567, 0.18634918642795975289557131741, 1.84192360072070073976371195044, 3.08922685911511800726255743696, 3.61358475823917058532946860367, 4.99262636771099118916621861472, 5.96649615572181673186670565505, 6.81474034523309764541307555146, 7.31278632568474624046090893972, 8.458493780205957967859883984360, 9.304285163120311695607075791229

Graph of the ZZ-function along the critical line