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Properties

Label 2-1368-1368.1051-c0-0-0
Degree 22
Conductor 13681368
Sign 0.188−0.982i0.188 - 0.982i
Analytic cond. 0.6827200.682720
Root an. cond. 0.8262690.826269
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  + (0.173 + 0.984i)2-s + 3-s + (−0.939 + 0.342i)4-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + 9-s + 1.53·11-s + (−0.939 + 0.342i)12-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + (0.173 + 0.984i)18-s + (−0.939 + 0.342i)19-s + (0.266 + 1.50i)22-s + (−0.5 − 0.866i)24-s + (0.173 − 0.984i)25-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + 3-s + (−0.939 + 0.342i)4-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + 9-s + 1.53·11-s + (−0.939 + 0.342i)12-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + (0.173 + 0.984i)18-s + (−0.939 + 0.342i)19-s + (0.266 + 1.50i)22-s + (−0.5 − 0.866i)24-s + (0.173 − 0.984i)25-s + ⋯

Functional equation

Λ(s)=(1368s/2 ΓC(s) L(s)=((0.188−0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1368s/2 ΓC(s) L(s)=((0.188−0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13681368    =    23⋅32⋅192^{3} \cdot 3^{2} \cdot 19
Sign: 0.188−0.982i0.188 - 0.982i
Analytic conductor: 0.6827200.682720
Root analytic conductor: 0.8262690.826269
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1368(1051,⋅)\chi_{1368} (1051, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1368, ( :0), 0.188−0.982i)(2,\ 1368,\ (\ :0),\ 0.188 - 0.982i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5597022921.559702292
L(12)L(\frac12) ≈\approx 1.5597022921.559702292
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+(−0.173−0.984i)T 1 + (-0.173 - 0.984i)T
3 1−T 1 - T
19 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
good5 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1−1.53T+T2 1 - 1.53T + T^{2}
13 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
17 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
23 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
29 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
31 1−T2 1 - T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.266−1.50i)T+(−0.939+0.342i)T2 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2}
43 1+(1.87+0.684i)T+(0.766+0.642i)T2 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2}
47 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
53 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
59 1+(0.326+0.118i)T+(0.766+0.642i)T2 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2}
61 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
67 1+(−0.266+1.50i)T+(−0.939−0.342i)T2 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2}
71 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
73 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
79 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
83 1+(0.173+0.300i)T+(−0.5+0.866i)T2 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2}
89 1+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}
97 1+(0.326+1.85i)T+(−0.939+0.342i)T2 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)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

−9.644588598169575563545602590437, −8.839251783025759231413925949064, −8.447611313288370694579297016318, −7.59626234976418489657337340151, −6.50705252398641487236156562425, −6.33090661288406209697110685151, −4.67394843388091085531201171298, −4.13177413080031401308217824498, −3.22134556471724146144083999877, −1.69916276353853756118770485195, 1.43845981839743621590915414958, 2.40723046801462471112230810726, 3.49145132306645658100101960287, 4.15604159493434402295917559932, 5.03280134109724917289783525895, 6.42330283254443637049414934348, 7.19383257656430740923727472997, 8.464115564058963581184869630547, 8.929169300938310182303125130055, 9.521618508322374557271220242217

Graph of the ZZ-function along the critical line