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Properties

Label 2-3267-27.14-c0-0-0
Degree 22
Conductor 32673267
Sign −0.396−0.918i-0.396 - 0.918i
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  + (−0.173 + 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.439 − 0.524i)5-s + (−0.939 − 0.342i)9-s + (−0.173 − 0.984i)12-s + (0.439 + 0.524i)15-s + (0.766 − 0.642i)16-s + (−0.233 + 0.642i)20-s + (0.592 + 1.62i)23-s + (0.0923 + 0.524i)25-s + (0.5 − 0.866i)27-s + (0.326 − 0.118i)31-s + 0.999·36-s + (−0.939 + 1.62i)37-s + (−0.592 + 0.342i)45-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.439 − 0.524i)5-s + (−0.939 − 0.342i)9-s + (−0.173 − 0.984i)12-s + (0.439 + 0.524i)15-s + (0.766 − 0.642i)16-s + (−0.233 + 0.642i)20-s + (0.592 + 1.62i)23-s + (0.0923 + 0.524i)25-s + (0.5 − 0.866i)27-s + (0.326 − 0.118i)31-s + 0.999·36-s + (−0.939 + 1.62i)37-s + (−0.592 + 0.342i)45-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((−0.396−0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((−0.396−0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: −0.396−0.918i-0.396 - 0.918i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(122,⋅)\chi_{3267} (122, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), −0.396−0.918i)(2,\ 3267,\ (\ :0),\ -0.396 - 0.918i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.82853325460.8285332546
L(12)L(\frac12) ≈\approx 0.82853325460.8285332546
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+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
11 1 1
good2 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
5 1+(−0.439+0.524i)T+(−0.173−0.984i)T2 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2}
7 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
13 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
23 1+(−0.592−1.62i)T+(−0.766+0.642i)T2 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2}
29 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
31 1+(−0.326+0.118i)T+(0.766−0.642i)T2 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2}
37 1+(0.939−1.62i)T+(−0.5−0.866i)T2 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2}
41 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
43 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
47 1+(0.439−1.20i)T+(−0.766−0.642i)T2 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2}
53 1−1.96iT−T2 1 - 1.96iT - T^{2}
59 1+(−1.26+1.50i)T+(−0.173−0.984i)T2 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2}
61 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
67 1+(0.0603−0.342i)T+(−0.939−0.342i)T2 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2}
71 1+(−1.11−0.642i)T+(0.5+0.866i)T2 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2}
73 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
79 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
83 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)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+(−1.17+0.984i)T+(0.173−0.984i)T2 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)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

−9.212309334705313436622465708168, −8.500649698285444552211723301284, −7.86025401553258196937696328288, −6.75289819809402636083907571815, −5.64785702612269900454509086233, −5.18324758556152947685935760475, −4.54345887630728799729863111730, −3.65019555839717160049966558737, −2.97473811417611200024454703847, −1.29875893146418546164645615957, 0.57480218064960379523749741278, 1.88656625822407242026446373898, 2.78007155965612752466602809950, 3.90255912381001307393038261701, 4.97214837081066650101141859059, 5.57235417199948006801225353451, 6.48940886095712393827728523943, 6.86700539859842131713615084433, 7.950286028258937349553921911758, 8.566205431770709856485264672456

Graph of the ZZ-function along the critical line