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Properties

Label 2-3267-99.52-c0-0-0
Degree 22
Conductor 32673267
Sign −0.509−0.860i-0.509 - 0.860i
Analytic cond. 1.630441.63044
Root an. cond. 1.276881.27688
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  + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)5-s + (0.913 − 0.406i)16-s + (−0.104 − 0.994i)20-s + (1 + 1.73i)23-s + (−0.913 − 0.406i)31-s + (−0.309 + 0.951i)37-s + (−0.978 − 0.207i)47-s + (−0.104 + 0.994i)49-s + (−0.809 + 0.587i)53-s + (−0.978 + 0.207i)59-s + (−0.809 + 0.587i)64-s + (0.5 + 0.866i)67-s + (−0.809 − 0.587i)71-s + (0.309 + 0.951i)80-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)5-s + (0.913 − 0.406i)16-s + (−0.104 − 0.994i)20-s + (1 + 1.73i)23-s + (−0.913 − 0.406i)31-s + (−0.309 + 0.951i)37-s + (−0.978 − 0.207i)47-s + (−0.104 + 0.994i)49-s + (−0.809 + 0.587i)53-s + (−0.978 + 0.207i)59-s + (−0.809 + 0.587i)64-s + (0.5 + 0.866i)67-s + (−0.809 − 0.587i)71-s + (0.309 + 0.951i)80-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((−0.509−0.860i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((−0.509−0.860i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: −0.509−0.860i-0.509 - 0.860i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(1207,⋅)\chi_{3267} (1207, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), −0.509−0.860i)(2,\ 3267,\ (\ :0),\ -0.509 - 0.860i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.74814950830.7481495083
L(12)L(\frac12) ≈\approx 0.74814950830.7481495083
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)
bad3 1 1
11 1 1
good2 1+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
5 1+(0.104−0.994i)T+(−0.978−0.207i)T2 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2}
7 1+(0.104−0.994i)T2 1 + (0.104 - 0.994i)T^{2}
13 1+(−0.669−0.743i)T2 1 + (-0.669 - 0.743i)T^{2}
17 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
19 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
23 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.104−0.994i)T2 1 + (0.104 - 0.994i)T^{2}
31 1+(0.913+0.406i)T+(0.669+0.743i)T2 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2}
37 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}
41 1+(0.104+0.994i)T2 1 + (0.104 + 0.994i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(0.978+0.207i)T+(0.913+0.406i)T2 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2}
53 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
59 1+(0.978−0.207i)T+(0.913−0.406i)T2 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2}
61 1+(−0.669+0.743i)T2 1 + (-0.669 + 0.743i)T^{2}
67 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
73 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
79 1+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
83 1+(−0.669+0.743i)T2 1 + (-0.669 + 0.743i)T^{2}
89 1+2T+T2 1 + 2T + T^{2}
97 1+(−0.104−0.994i)T+(−0.978+0.207i)T2 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)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

−9.202606488734546123972439270977, −8.308142820759727689462562170620, −7.54966288156008324667718041845, −7.01254372397019116801852401184, −6.01474678339651415569891320995, −5.24132896100674764644662322412, −4.42468690809598752094662416111, −3.43077939147794961493488788904, −2.97825849042611852837872480355, −1.45490200566881147822610228346, 0.49757716454336614095278597182, 1.67144306035248999820036698691, 3.09851753439624664637011447344, 4.09163191980560521742651966030, 4.82109951353587343163821252300, 5.24496523306584725108749674603, 6.22283269815819783694168032239, 7.12552822275765368173513627696, 8.136548456855850905701658031250, 8.676325850053130464016220119829

Graph of the ZZ-function along the critical line