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Properties

Label 2-3267-11.2-c0-0-1
Degree 22
Conductor 32673267
Sign 0.681+0.731i0.681 + 0.731i
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)=((0.681+0.731i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((0.681+0.731i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: 0.681+0.731i0.681 + 0.731i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(838,⋅)\chi_{3267} (838, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), 0.681+0.731i)(2,\ 3267,\ (\ :0),\ 0.681 + 0.731i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1336991431.133699143
L(12)L(\frac12) ≈\approx 1.1336991431.133699143
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

−8.600318455223118400348272185968, −8.437462614021968044226496234106, −7.04515517372000576502886756315, −6.51690755325409044029723532081, −5.65650055307991812059617099624, −4.80831694470678973350776581739, −4.16858829870369060677789181889, −3.37066188118227708094761357023, −1.89995998755732687185511280733, −0.886188653004514832453301012965, 1.16619913820001205251355152875, 2.51323951846356027015164846669, 3.68462016643140450619198518862, 3.99878968579795002560419198531, 5.18830002093268639229146036758, 5.79209469452651247114826755125, 6.61353958824165358962593857295, 7.74723202868378783724316144784, 8.300577029821686795597087425657, 8.696172754021391535641798082821

Graph of the ZZ-function along the critical line