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Properties

Label 2-3267-297.146-c0-0-0
Degree 22
Conductor 32673267
Sign 0.568+0.822i0.568 + 0.822i
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  + (−0.848 − 0.529i)3-s + (0.990 + 0.139i)4-s + (1.63 − 1.10i)5-s + (0.438 + 0.898i)9-s + (−0.766 − 0.642i)12-s + (−1.96 + 0.0687i)15-s + (0.961 + 0.275i)16-s + (1.77 − 0.863i)20-s + (−1.70 + 0.300i)23-s + (1.07 − 2.66i)25-s + (0.104 − 0.994i)27-s + (1.10 + 1.06i)31-s + (0.309 + 0.951i)36-s + (0.339 − 0.0722i)37-s + (1.70 + 0.984i)45-s + ⋯
L(s)  = 1  + (−0.848 − 0.529i)3-s + (0.990 + 0.139i)4-s + (1.63 − 1.10i)5-s + (0.438 + 0.898i)9-s + (−0.766 − 0.642i)12-s + (−1.96 + 0.0687i)15-s + (0.961 + 0.275i)16-s + (1.77 − 0.863i)20-s + (−1.70 + 0.300i)23-s + (1.07 − 2.66i)25-s + (0.104 − 0.994i)27-s + (1.10 + 1.06i)31-s + (0.309 + 0.951i)36-s + (0.339 − 0.0722i)37-s + (1.70 + 0.984i)45-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((0.568+0.822i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((0.568+0.822i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: 0.568+0.822i0.568 + 0.822i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(1334,⋅)\chi_{3267} (1334, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), 0.568+0.822i)(2,\ 3267,\ (\ :0),\ 0.568 + 0.822i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6566604951.656660495
L(12)L(\frac12) ≈\approx 1.6566604951.656660495
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+(0.848+0.529i)T 1 + (0.848 + 0.529i)T
11 1 1
good2 1+(−0.990−0.139i)T2 1 + (-0.990 - 0.139i)T^{2}
5 1+(−1.63+1.10i)T+(0.374−0.927i)T2 1 + (-1.63 + 1.10i)T + (0.374 - 0.927i)T^{2}
7 1+(0.559+0.829i)T2 1 + (0.559 + 0.829i)T^{2}
13 1+(−0.882−0.469i)T2 1 + (-0.882 - 0.469i)T^{2}
17 1+(0.978+0.207i)T2 1 + (0.978 + 0.207i)T^{2}
19 1+(0.913+0.406i)T2 1 + (0.913 + 0.406i)T^{2}
23 1+(1.70−0.300i)T+(0.939−0.342i)T2 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2}
29 1+(−0.438+0.898i)T2 1 + (-0.438 + 0.898i)T^{2}
31 1+(−1.10−1.06i)T+(0.0348+0.999i)T2 1 + (-1.10 - 1.06i)T + (0.0348 + 0.999i)T^{2}
37 1+(−0.339+0.0722i)T+(0.913−0.406i)T2 1 + (-0.339 + 0.0722i)T + (0.913 - 0.406i)T^{2}
41 1+(−0.438−0.898i)T2 1 + (-0.438 - 0.898i)T^{2}
43 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
47 1+(−0.0952−0.677i)T+(−0.961+0.275i)T2 1 + (-0.0952 - 0.677i)T + (-0.961 + 0.275i)T^{2}
53 1+(0.755+1.04i)T+(−0.309+0.951i)T2 1 + (0.755 + 1.04i)T + (-0.309 + 0.951i)T^{2}
59 1+(−1.01−0.791i)T+(0.241+0.970i)T2 1 + (-1.01 - 0.791i)T + (0.241 + 0.970i)T^{2}
61 1+(0.0348−0.999i)T2 1 + (0.0348 - 0.999i)T^{2}
67 1+(1.17−0.984i)T+(0.173−0.984i)T2 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2}
71 1+(0.680+0.0715i)T+(0.978+0.207i)T2 1 + (0.680 + 0.0715i)T + (0.978 + 0.207i)T^{2}
73 1+(−0.104+0.994i)T2 1 + (-0.104 + 0.994i)T^{2}
79 1+(0.990+0.139i)T2 1 + (0.990 + 0.139i)T^{2}
83 1+(0.882−0.469i)T2 1 + (0.882 - 0.469i)T^{2}
89 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
97 1+(1.05−1.55i)T+(−0.374−0.927i)T2 1 + (1.05 - 1.55i)T + (-0.374 - 0.927i)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

−8.545905002478211210567583561881, −8.003731634421338418731652107880, −6.96064248613560689157784830430, −6.31201910901425374717134521265, −5.81262365722620490783292739717, −5.21257525315089186326933393298, −4.29645924801665380740766312984, −2.69507873007885707706161528818, −1.86606113241617415364915335791, −1.23612967333083022947580551824, 1.52607123399347225913304649564, 2.40262458341313793041152527560, 3.20307889884968796085697716315, 4.38366274961087479552819070170, 5.56580220840992990309777959413, 5.98496456961289850757427377190, 6.45756700865933892647559977038, 7.09166011717856809910002653088, 8.050913954674918186660609041387, 9.410053394750038362495143994908

Graph of the ZZ-function along the critical line