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Properties

Label 2-1872-117.70-c0-0-1
Degree 22
Conductor 18721872
Sign 0.800−0.599i0.800 - 0.599i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
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  + (0.5 + 0.866i)3-s + (1.36 − 0.366i)7-s + (−0.499 + 0.866i)9-s + (1.36 − 0.366i)11-s + (−0.866 + 0.5i)13-s − i·17-s + (1 − i)19-s + (1 + 0.999i)21-s + (−0.866 + 0.5i)23-s + (−0.866 − 0.5i)25-s − 0.999·27-s + (1 + 0.999i)33-s + (1 + i)37-s + (−0.866 − 0.499i)39-s + (−1.36 − 0.366i)41-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (1.36 − 0.366i)7-s + (−0.499 + 0.866i)9-s + (1.36 − 0.366i)11-s + (−0.866 + 0.5i)13-s − i·17-s + (1 − i)19-s + (1 + 0.999i)21-s + (−0.866 + 0.5i)23-s + (−0.866 − 0.5i)25-s − 0.999·27-s + (1 + 0.999i)33-s + (1 + i)37-s + (−0.866 − 0.499i)39-s + (−1.36 − 0.366i)41-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((0.800−0.599i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s) L(s)=((0.800−0.599i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 0.800−0.599i0.800 - 0.599i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1872(1825,⋅)\chi_{1872} (1825, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :0), 0.800−0.599i)(2,\ 1872,\ (\ :0),\ 0.800 - 0.599i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5813149501.581314950
L(12)L(\frac12) ≈\approx 1.5813149501.581314950
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
good5 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
7 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
11 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
23 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
29 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
31 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
37 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
41 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
43 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
47 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
53 1+T+T2 1 + T + T^{2}
59 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
61 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
71 1−iT2 1 - iT^{2}
73 1+iT2 1 + iT^{2}
79 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
89 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
97 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)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

−9.537180488329848088998119403018, −8.845724661098549185477509225712, −7.938212589205832580112158427807, −7.41184528901657582291063166299, −6.31281513578766730257923791513, −5.07938026219760773772115606422, −4.63634961670779148986093213389, −3.80963058761701487157235716757, −2.70618525650891556390506239541, −1.52421086606574597006503406665, 1.50667707705467885057641092283, 2.02776429859673837017409988876, 3.42482540758769838936075216008, 4.31482436325630064132513742793, 5.45644137040267498218810802556, 6.18076058543716551068180160290, 7.14061249417357075701763428943, 8.026477024728232844472920215938, 8.169956766363443626170422291972, 9.308089923509075376140031930325

Graph of the ZZ-function along the critical line