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Properties

Label 2-2736-228.227-c0-0-8
Degree 22
Conductor 27362736
Sign 0.0917+0.995i0.0917 + 0.995i
Analytic cond. 1.365441.36544
Root an. cond. 1.168521.16852
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.517i·5-s − i·7-s + 0.517·11-s − 1.93i·17-s + i·19-s − 1.41·23-s + 0.732·25-s − 0.517·35-s − 1.73i·43-s − 1.93·47-s − 0.267i·55-s + 1.73·61-s − 73-s − 0.517i·77-s + 1.41·83-s + ⋯
L(s)  = 1  − 0.517i·5-s − i·7-s + 0.517·11-s − 1.93i·17-s + i·19-s − 1.41·23-s + 0.732·25-s − 0.517·35-s − 1.73i·43-s − 1.93·47-s − 0.267i·55-s + 1.73·61-s − 73-s − 0.517i·77-s + 1.41·83-s + ⋯

Functional equation

Λ(s)=(2736s/2 ΓC(s) L(s)=((0.0917+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2736s/2 ΓC(s) L(s)=((0.0917+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27362736    =    24⋅32⋅192^{4} \cdot 3^{2} \cdot 19
Sign: 0.0917+0.995i0.0917 + 0.995i
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2736(2735,⋅)\chi_{2736} (2735, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2736, ( :0), 0.0917+0.995i)(2,\ 2736,\ (\ :0),\ 0.0917 + 0.995i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1423580041.142358004
L(12)L(\frac12) ≈\approx 1.1423580041.142358004
L(1)L(1) 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
19 1−iT 1 - iT
good5 1+0.517iT−T2 1 + 0.517iT - T^{2}
7 1+iT−T2 1 + iT - T^{2}
11 1−0.517T+T2 1 - 0.517T + T^{2}
13 1−T2 1 - T^{2}
17 1+1.93iT−T2 1 + 1.93iT - T^{2}
23 1+1.41T+T2 1 + 1.41T + T^{2}
29 1+T2 1 + T^{2}
31 1+T2 1 + T^{2}
37 1−T2 1 - T^{2}
41 1+T2 1 + T^{2}
43 1+1.73iT−T2 1 + 1.73iT - T^{2}
47 1+1.93T+T2 1 + 1.93T + T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−1.73T+T2 1 - 1.73T + T^{2}
67 1+T2 1 + T^{2}
71 1−T2 1 - T^{2}
73 1+T+T2 1 + T + T^{2}
79 1+T2 1 + T^{2}
83 1−1.41T+T2 1 - 1.41T + T^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - T^{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

−8.802613125151395881222116633159, −8.077710667137686646634459608322, −7.27117413827582439974439281521, −6.71218471243661975243681602585, −5.67482753727372877335976260354, −4.85856662747130656477379688143, −4.08997092199353151072725277805, −3.29171146360642447334460619284, −1.96585402786050716035646503160, −0.75260675886657805150076381824, 1.63991485043357070971683986163, 2.57390535570138044752414877232, 3.53268164377710775151331755105, 4.42295179270097513805726384324, 5.42986620025226661133113198453, 6.30821089635726170146844717923, 6.61331032585000722876570364095, 7.84635489511807641632945970701, 8.420941329034339995430898391752, 9.117420526265624875008469758275

Graph of the ZZ-function along the critical line