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Properties

Label 2-3864-3864.125-c0-0-2
Degree 22
Conductor 38643864
Sign 0.325−0.945i0.325 - 0.945i
Analytic cond. 1.928381.92838
Root an. cond. 1.388661.38866
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  + (−0.755 + 0.654i)2-s + (0.707 − 0.707i)3-s + (0.142 − 0.989i)4-s + (−0.729 + 1.59i)5-s + (−0.0713 + 0.997i)6-s + (−0.281 + 0.959i)7-s + (0.540 + 0.841i)8-s − 1.00i·9-s + (−0.494 − 1.68i)10-s + (−0.599 − 0.800i)12-s + (1.91 − 0.562i)13-s + (−0.415 − 0.909i)14-s + (0.613 + 1.64i)15-s + (−0.959 − 0.281i)16-s + (0.654 + 0.755i)18-s + (0.691 + 0.0994i)19-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)2-s + (0.707 − 0.707i)3-s + (0.142 − 0.989i)4-s + (−0.729 + 1.59i)5-s + (−0.0713 + 0.997i)6-s + (−0.281 + 0.959i)7-s + (0.540 + 0.841i)8-s − 1.00i·9-s + (−0.494 − 1.68i)10-s + (−0.599 − 0.800i)12-s + (1.91 − 0.562i)13-s + (−0.415 − 0.909i)14-s + (0.613 + 1.64i)15-s + (−0.959 − 0.281i)16-s + (0.654 + 0.755i)18-s + (0.691 + 0.0994i)19-s + ⋯

Functional equation

Λ(s)=(3864s/2 ΓC(s) L(s)=((0.325−0.945i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3864s/2 ΓC(s) L(s)=((0.325−0.945i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38643864    =    23⋅3⋅7⋅232^{3} \cdot 3 \cdot 7 \cdot 23
Sign: 0.325−0.945i0.325 - 0.945i
Analytic conductor: 1.928381.92838
Root analytic conductor: 1.388661.38866
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3864(125,⋅)\chi_{3864} (125, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3864, ( :0), 0.325−0.945i)(2,\ 3864,\ (\ :0),\ 0.325 - 0.945i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0633312671.063331267
L(12)L(\frac12) ≈\approx 1.0633312671.063331267
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+(0.755−0.654i)T 1 + (0.755 - 0.654i)T
3 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
7 1+(0.281−0.959i)T 1 + (0.281 - 0.959i)T
23 1+(−0.959+0.281i)T 1 + (-0.959 + 0.281i)T
good5 1+(0.729−1.59i)T+(−0.654−0.755i)T2 1 + (0.729 - 1.59i)T + (-0.654 - 0.755i)T^{2}
11 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
13 1+(−1.91+0.562i)T+(0.841−0.540i)T2 1 + (-1.91 + 0.562i)T + (0.841 - 0.540i)T^{2}
17 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
19 1+(−0.691−0.0994i)T+(0.959+0.281i)T2 1 + (-0.691 - 0.0994i)T + (0.959 + 0.281i)T^{2}
29 1+(−0.959+0.281i)T2 1 + (-0.959 + 0.281i)T^{2}
31 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
37 1+(−0.654+0.755i)T2 1 + (-0.654 + 0.755i)T^{2}
41 1+(−0.654−0.755i)T2 1 + (-0.654 - 0.755i)T^{2}
43 1+(0.415+0.909i)T2 1 + (0.415 + 0.909i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
59 1+(−0.527−1.79i)T+(−0.841+0.540i)T2 1 + (-0.527 - 1.79i)T + (-0.841 + 0.540i)T^{2}
61 1+(−0.764−1.18i)T+(−0.415+0.909i)T2 1 + (-0.764 - 1.18i)T + (-0.415 + 0.909i)T^{2}
67 1+(−0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
71 1+(0.989−0.857i)T+(0.142−0.989i)T2 1 + (0.989 - 0.857i)T + (0.142 - 0.989i)T^{2}
73 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
79 1+(−0.0801−0.273i)T+(−0.841+0.540i)T2 1 + (-0.0801 - 0.273i)T + (-0.841 + 0.540i)T^{2}
83 1+(−0.665−1.45i)T+(−0.654+0.755i)T2 1 + (-0.665 - 1.45i)T + (-0.654 + 0.755i)T^{2}
89 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
97 1+(−0.654−0.755i)T2 1 + (-0.654 - 0.755i)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

−8.559894019348456703241786614791, −8.131680235477783488126985256046, −7.29662815509123612043502226223, −6.82522627264108946366304744444, −6.13461864280989785900533629938, −5.59440435344948523781351796603, −3.93767779753633820086109061461, −3.09379343540250470182153509782, −2.51317598655010363971402702464, −1.18512347596909978399004368816, 0.890431864540458737609533230502, 1.70600558628390303536954715121, 3.36216211398992226025699885641, 3.68221128545968511634189451459, 4.42357237153044756064195765846, 5.11729784007815075888221702349, 6.51334136838273299987697701294, 7.56827537349447428761459558437, 8.006075979094586508914777545584, 8.792076938573322869586056108202

Graph of the ZZ-function along the critical line