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Properties

Label 2-2736-228.227-c0-0-6
Degree 22
Conductor 27362736
Sign 0.908+0.418i0.908 + 0.418i
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)=((0.908+0.418i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2736s/2 ΓC(s) L(s)=((0.908+0.418i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27362736    =    24⋅32⋅192^{4} \cdot 3^{2} \cdot 19
Sign: 0.908+0.418i0.908 + 0.418i
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), 0.908+0.418i)(2,\ 2736,\ (\ :0),\ 0.908 + 0.418i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2106647591.210664759
L(12)L(\frac12) ≈\approx 1.2106647591.210664759
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

−9.096008575240992245813090231235, −8.375645413996755458343302692788, −7.34642154618976860925397957617, −6.82796769403867502820459119278, −5.64369388777373917597613248924, −5.09970589649366700071149069247, −4.48916422537915226915426061117, −2.92106252210829945321225105600, −2.57234638897194336547209514563, −0.934542519872865180945627982580, 1.21864831734537424724549754887, 2.45359674966082818640467731377, 3.59128030475921440451526726166, 4.09544319091030015399388171934, 5.24863933978751934757777936689, 6.02868542648641723130892709157, 6.93441487770941152627669203106, 7.41585331419030246464632446096, 8.309961467227838114646638501445, 8.914225315626013043074031947496

Graph of the ZZ-function along the critical line