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Properties

Label 2-3549-273.122-c0-0-3
Degree 22
Conductor 35493549
Sign 0.763−0.645i0.763 - 0.645i
Analytic cond. 1.771181.77118
Root an. cond. 1.330851.33085
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  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)4-s − i·7-s + (0.499 + 0.866i)9-s + (0.499 + 0.866i)12-s + (0.499 + 0.866i)16-s + (−0.133 + 0.5i)19-s + (0.5 − 0.866i)21-s + (−0.866 − 0.5i)25-s + 0.999i·27-s + (0.5 − 0.866i)28-s + (1.36 − 0.366i)31-s + 0.999i·36-s + (−0.5 + 1.86i)37-s − 1.73i·43-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)4-s − i·7-s + (0.499 + 0.866i)9-s + (0.499 + 0.866i)12-s + (0.499 + 0.866i)16-s + (−0.133 + 0.5i)19-s + (0.5 − 0.866i)21-s + (−0.866 − 0.5i)25-s + 0.999i·27-s + (0.5 − 0.866i)28-s + (1.36 − 0.366i)31-s + 0.999i·36-s + (−0.5 + 1.86i)37-s − 1.73i·43-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((0.763−0.645i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((0.763−0.645i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: 0.763−0.645i0.763 - 0.645i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(1760,⋅)\chi_{3549} (1760, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), 0.763−0.645i)(2,\ 3549,\ (\ :0),\ 0.763 - 0.645i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1569477282.156947728
L(12)L(\frac12) ≈\approx 2.1569477282.156947728
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
7 1+iT 1 + iT
13 1 1
good2 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
5 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
11 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(0.133−0.5i)T+(−0.866−0.5i)T2 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2}
23 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
37 1+(0.5−1.86i)T+(−0.866−0.5i)T2 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2}
41 1−iT2 1 - iT^{2}
43 1+1.73iT−T2 1 + 1.73iT - T^{2}
47 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
61 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
67 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
71 1+iT2 1 + iT^{2}
73 1+(0.5+1.86i)T+(−0.866+0.5i)T2 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2}
79 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
83 1−iT2 1 - iT^{2}
89 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
97 1+(1.36+1.36i)T+iT2 1 + (1.36 + 1.36i)T + iT^{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

−8.467482654682537256233776909777, −8.224515417755292801547829713750, −7.39800053113114581642090763855, −6.84243264972210066045982469751, −5.96960983158638657413887327335, −4.76939617025721734329560526040, −3.99492278977204482655334756746, −3.38069923798820802894288882083, −2.49807628446436336015765398928, −1.54132167699311601105641325056, 1.29292888654026365294802064565, 2.28374824693824292094815139105, 2.76652666260140134675378462976, 3.78752850608061866641817781884, 4.99699419022999316002492966693, 5.86601560726842506765778402557, 6.47348845057185527055172146407, 7.18980379388200915578803852445, 7.927650670350366899283112586007, 8.586625890859466504272731828197

Graph of the ZZ-function along the critical line