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Properties

Label 2-30e2-180.83-c0-0-1
Degree $2$
Conductor $900$
Sign $-0.619 + 0.784i$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.619 + 0.784i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ -0.619 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9947326415\)
\(L(\frac12)\) \(\approx\) \(0.9947326415\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
good7 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line