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Properties

Label 2-30e2-180.83-c0-0-1
Degree 22
Conductor 900900
Sign −0.619+0.784i-0.619 + 0.784i
Analytic cond. 0.4491580.449158
Root an. cond. 0.6701920.670192
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  + (−0.258 − 0.965i)2-s + (0.258 − 0.965i)3-s + (−0.866 + 0.499i)4-s − 6-s + (1.67 − 0.448i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)12-s + (−0.866 − 1.50i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)18-s − 1.73i·21-s + (0.258 − 0.965i)23-s + (0.866 − 0.5i)24-s + (−0.707 + 0.707i)27-s + (−1.22 + 1.22i)28-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.258 − 0.965i)3-s + (−0.866 + 0.499i)4-s − 6-s + (1.67 − 0.448i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)12-s + (−0.866 − 1.50i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)18-s − 1.73i·21-s + (0.258 − 0.965i)23-s + (0.866 − 0.5i)24-s + (−0.707 + 0.707i)27-s + (−1.22 + 1.22i)28-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((−0.619+0.784i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((−0.619+0.784i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: −0.619+0.784i-0.619 + 0.784i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(443,⋅)\chi_{900} (443, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), −0.619+0.784i)(2,\ 900,\ (\ :0),\ -0.619 + 0.784i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.99473264150.9947326415
L(12)L(\frac12) ≈\approx 0.99473264150.9947326415
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)
bad2 1+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
3 1+(−0.258+0.965i)T 1 + (-0.258 + 0.965i)T
5 1 1
good7 1+(−1.67+0.448i)T+(0.866−0.5i)T2 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
13 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
17 1+iT2 1 + iT^{2}
19 1+T2 1 + T^{2}
23 1+(−0.258+0.965i)T+(−0.866−0.5i)T2 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}
29 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1−iT2 1 - iT^{2}
41 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
43 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
47 1+(0.258+0.965i)T+(−0.866+0.5i)T2 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2}
53 1−iT2 1 - iT^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.448−1.67i)T+(−0.866−0.5i)T2 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2}
71 1+T2 1 + T^{2}
73 1+iT2 1 + iT^{2}
79 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
83 1+(−0.965+0.258i)T+(0.866−0.5i)T2 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2}
89 1−1.73T+T2 1 - 1.73T + T^{2}
97 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)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

−10.24769090655272555788383931732, −8.934600161427883444817444426501, −8.447606816627508189767210083192, −7.69561770967871707850124330363, −6.93077021946505671280480442735, −5.40739762224978295849964234291, −4.56676315763756468429961677014, −3.34919362524968072177451099687, −2.09737856918546255331902039154, −1.25355587885700000831095309657, 1.91623440393042965809613716230, 3.68245633247113037934359810588, 4.72622105173765869168073841621, 5.24293942249888439948333190860, 6.13099296937079081101371289858, 7.62022865433085171003735629871, 8.028713959023576754895725418429, 8.910079631423548087992266442669, 9.492387051047927492376550481426, 10.47755778289601806030086640876

Graph of the ZZ-function along the critical line