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Properties

Label 2-462-33.8-c1-0-8
Degree $2$
Conductor $462$
Sign $0.491 + 0.871i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (−1.31 − 1.12i)3-s + (0.309 − 0.951i)4-s + (−1.88 + 2.60i)5-s + (1.72 + 0.132i)6-s + (−0.951 − 0.309i)7-s + (0.309 + 0.951i)8-s + (0.479 + 2.96i)9-s − 3.21i·10-s + (0.355 − 3.29i)11-s + (−1.47 + 0.907i)12-s + (1.19 + 1.64i)13-s + (0.951 − 0.309i)14-s + (5.41 − 1.30i)15-s + (−0.809 − 0.587i)16-s + (−1.57 − 1.14i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.761 − 0.648i)3-s + (0.154 − 0.475i)4-s + (−0.844 + 1.16i)5-s + (0.705 + 0.0542i)6-s + (−0.359 − 0.116i)7-s + (0.109 + 0.336i)8-s + (0.159 + 0.987i)9-s − 1.01i·10-s + (0.107 − 0.994i)11-s + (−0.425 + 0.261i)12-s + (0.332 + 0.457i)13-s + (0.254 − 0.0825i)14-s + (1.39 − 0.338i)15-s + (−0.202 − 0.146i)16-s + (−0.382 − 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.491 + 0.871i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.491 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.422322 - 0.246664i\)
\(L(\frac12)\) \(\approx\) \(0.422322 - 0.246664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (1.31 + 1.12i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.355 + 3.29i)T \)
good5 \( 1 + (1.88 - 2.60i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.19 - 1.64i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.57 + 1.14i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.14 + 0.698i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.70iT - 23T^{2} \)
29 \( 1 + (-0.422 + 1.29i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.07 + 0.779i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.29 + 10.1i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.23 + 3.80i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.48iT - 43T^{2} \)
47 \( 1 + (3.88 - 1.26i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.85 - 5.30i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-11.7 - 3.82i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.27 + 3.13i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.77T + 67T^{2} \)
71 \( 1 + (-3.19 + 4.39i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.22 - 1.37i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.0 - 13.8i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (13.6 + 9.93i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.34iT - 89T^{2} \)
97 \( 1 + (8.66 - 6.29i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.99079019989649920279143741021, −10.28172710430308621292639951783, −8.913848113486614850238308464894, −7.957656510785249508539982689217, −7.04970508337644750646736007061, −6.57127026933422906056924926697, −5.60830802326071992323412637404, −4.02980525048705706549304280447, −2.56102573306858163921624091264, −0.47293263131551119818163585770, 1.19022896441923925965752605718, 3.41018297790384144259862058007, 4.40811471114655875702318136306, 5.27527399270137232469090607100, 6.60615268691749002614135802026, 7.78015850925635512261542062002, 8.667381266354079938445264121593, 9.602669951743158553426815310739, 10.12002303100565305725470816034, 11.42249801035732842525282072714

Graph of the $Z$-function along the critical line