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Properties

Label 2-462-11.9-c1-0-9
Degree $2$
Conductor $462$
Sign $0.780 + 0.625i$
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  + (−0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.190 − 0.587i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 0.618·10-s + (0.309 − 3.30i)11-s − 0.999·12-s + (0.618 − 1.90i)13-s + (0.809 − 0.587i)14-s + (−0.5 − 0.363i)15-s + (0.309 + 0.951i)16-s + (0.118 + 0.363i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.0854 − 0.262i)5-s + (0.126 + 0.388i)6-s + (−0.305 − 0.222i)7-s + (0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + 0.195·10-s + (0.0931 − 0.995i)11-s − 0.288·12-s + (0.171 − 0.527i)13-s + (0.216 − 0.157i)14-s + (−0.129 − 0.0937i)15-s + (0.0772 + 0.237i)16-s + (0.0286 + 0.0881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.780 + 0.625i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.780 + 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20703 - 0.424168i\)
\(L(\frac12)\) \(\approx\) \(1.20703 - 0.424168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.309 + 3.30i)T \)
good5 \( 1 + (0.190 + 0.587i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.618 + 1.90i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.118 - 0.363i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.92 + 2.85i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.85T + 23T^{2} \)
29 \( 1 + (-2 - 1.45i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.42 + 4.39i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.11 + 2.26i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.11 - 1.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (5.23 - 3.80i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.145 + 0.449i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.23 - 3.07i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.76 - 11.5i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + (-2 - 6.15i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.23 - 5.98i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.47 + 13.7i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.854 - 2.62i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (3.14 - 9.68i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.81845967546012798677513800831, −9.851738572496570774815697777705, −8.920116489577694913799819948704, −8.224547994894713332508401785068, −7.38927428209024765781830184714, −6.37371731216958144508431610859, −5.48738064838559483134581045076, −4.12107557411252908726188532367, −2.88855135146281238619127113332, −0.864116749979770617001202473838, 1.80471619089478651305007458642, 3.07554581297700805941662662986, 4.07167411250115941029182570773, 5.19322122455632027597117591744, 6.66147855206499859122448230872, 7.66596569593905994692533182505, 8.648509276989708205052331934261, 9.585131607825182983609236622696, 10.07186709281568603037880450242, 11.06032535279543966219976225360

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