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Properties

Label 2-462-231.32-c1-0-6
Degree $2$
Conductor $462$
Sign $-0.764 - 0.645i$
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  + (0.5 + 0.866i)2-s + (−1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1.13 + 0.656i)5-s − 1.73i·6-s + (1.13 − 2.38i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−1.13 − 0.656i)10-s + (−0.637 + 3.25i)11-s + (1.49 − 0.866i)12-s + 2.62i·13-s + (2.63 − 0.209i)14-s + 2.27·15-s + (−0.5 − 0.866i)16-s + (−1.63 + 2.83i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.508 + 0.293i)5-s − 0.707i·6-s + (0.429 − 0.902i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.359 − 0.207i)10-s + (−0.192 + 0.981i)11-s + (0.433 − 0.250i)12-s + 0.728i·13-s + (0.704 − 0.0559i)14-s + 0.587·15-s + (−0.125 − 0.216i)16-s + (−0.397 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.764 - 0.645i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.764 - 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244794 + 0.669388i\)
\(L(\frac12)\) \(\approx\) \(0.244794 + 0.669388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-1.13 + 2.38i)T \)
11 \( 1 + (0.637 - 3.25i)T \)
good5 \( 1 + (1.13 - 0.656i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 2.62iT - 13T^{2} \)
17 \( 1 + (1.63 - 2.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.63 - 3.25i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.91 - 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-5.13 + 8.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.63 - 6.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.54T + 41T^{2} \)
43 \( 1 - 9.97iT - 43T^{2} \)
47 \( 1 + (-1.91 + 1.10i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.6 - 6.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 0.866i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.27 + 7.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.97iT - 71T^{2} \)
73 \( 1 + (4.54 + 2.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.86 - 1.07i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-2.27 + 1.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−11.57476171264606106767165246285, −10.63247952078585089533220627734, −9.810057228517895365079361589802, −8.128799442971747533984204538327, −7.64243499821712518522144808683, −6.71718815558134730133236770552, −6.00725779775747951782304529813, −4.54965996642917334893247881179, −4.09686518319191644436420937094, −1.86754804333366106405253969971, 0.43024360843908035024280316747, 2.53458499637883610415310211941, 3.93671163128592824140193284173, 4.90443302478680409877222450870, 5.67043403645455554951024237999, 6.65780184701942052251785932214, 8.352909856698684112842899510310, 8.863874374057283954921518016519, 10.20153364937140569515037438091, 10.80333189017125998752661703557

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