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Properties

Label 2-3267-11.6-c0-0-3
Degree 22
Conductor 32673267
Sign 0.292+0.956i0.292 + 0.956i
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  + (−0.809 + 0.587i)4-s + (−0.304 − 0.418i)7-s + (−1.83 + 0.596i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (0.492 + 0.159i)28-s + (−0.535 − 1.64i)31-s − 1.41i·43-s + (0.226 − 0.696i)49-s + (1.13 − 1.56i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s + 1.73·67-s + (−1.13 − 1.56i)73-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)4-s + (−0.304 − 0.418i)7-s + (−1.83 + 0.596i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (0.492 + 0.159i)28-s + (−0.535 − 1.64i)31-s − 1.41i·43-s + (0.226 − 0.696i)49-s + (1.13 − 1.56i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s + 1.73·67-s + (−1.13 − 1.56i)73-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((0.292+0.956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((0.292+0.956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: 0.292+0.956i0.292 + 0.956i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(2998,⋅)\chi_{3267} (2998, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), 0.292+0.956i)(2,\ 3267,\ (\ :0),\ 0.292 + 0.956i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.64417235380.6441723538
L(12)L(\frac12) ≈\approx 0.64417235380.6441723538
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+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
5 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
7 1+(0.304+0.418i)T+(−0.309+0.951i)T2 1 + (0.304 + 0.418i)T + (-0.309 + 0.951i)T^{2}
13 1+(1.83−0.596i)T+(0.809−0.587i)T2 1 + (1.83 - 0.596i)T + (0.809 - 0.587i)T^{2}
17 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
19 1+(−0.831+1.14i)T+(−0.309−0.951i)T2 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2}
23 1+T2 1 + T^{2}
29 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
31 1+(0.535+1.64i)T+(−0.809+0.587i)T2 1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2}
37 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
41 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
43 1+1.41iT−T2 1 + 1.41iT - T^{2}
47 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
53 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
59 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
61 1+(1.34+0.437i)T+(0.809+0.587i)T2 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2}
67 1−1.73T+T2 1 - 1.73T + T^{2}
71 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
73 1+(1.13+1.56i)T+(−0.309+0.951i)T2 1 + (1.13 + 1.56i)T + (-0.309 + 0.951i)T^{2}
79 1+(−1.83+0.596i)T+(0.809−0.587i)T2 1 + (-1.83 + 0.596i)T + (0.809 - 0.587i)T^{2}
83 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
89 1+T2 1 + T^{2}
97 1+(−0.309−0.951i)T+(−0.809+0.587i)T2 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)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

−8.869676825760762830205283667505, −7.72256321691022467921170622070, −7.35683961957015230975891221853, −6.66676201213035514869613039975, −5.33131276202724699023675114036, −4.84887751292469121028238542200, −4.03273756133781538159435564945, −3.15197777062040201778723862841, −2.23750292159158818923295133690, −0.43068129681576193715264096459, 1.22554977726817142538065734547, 2.55117617851007652666873958086, 3.42251186699316920350359868858, 4.57340288780439488799194435188, 5.15921035436385732826333329337, 5.77369127872901694884807182199, 6.69492827332307969169185530593, 7.58087061093774244561794553613, 8.274175481943559679583579421570, 9.108693262732714004245426640441

Graph of the ZZ-function along the critical line