Properties

Label 2-462-11.3-c1-0-5
Degree $2$
Conductor $462$
Sign $0.642 + 0.766i$
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 + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.224i)5-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 0.381·10-s + (3.04 − 1.31i)11-s − 12-s + (−1 + 0.726i)13-s + (0.309 + 0.951i)14-s + (0.118 − 0.363i)15-s + (−0.809 − 0.587i)16-s + (2.11 + 1.53i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.138 + 0.100i)5-s + (0.330 + 0.239i)6-s + (0.116 − 0.359i)7-s + (0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s − 0.120·10-s + (0.918 − 0.396i)11-s − 0.288·12-s + (−0.277 + 0.201i)13-s + (0.0825 + 0.254i)14-s + (0.0304 − 0.0937i)15-s + (−0.202 − 0.146i)16-s + (0.513 + 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915872 - 0.427234i\)
\(L(\frac12)\) \(\approx\) \(0.915872 - 0.427234i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-3.04 + 1.31i)T \)
good5 \( 1 + (-0.309 - 0.224i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (1 - 0.726i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.11 - 1.53i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.42 + 7.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.38T + 23T^{2} \)
29 \( 1 + (0.145 - 0.449i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.92 + 2.85i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.5 + 4.61i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.35 + 4.16i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.23T + 43T^{2} \)
47 \( 1 + (-0.145 - 0.449i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.85 + 5.70i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.854 - 2.62i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-9.70 - 7.05i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.61 - 11.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.85 - 4.97i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.38 - 1.73i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (0.381 - 0.277i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.96595307576124945593833762686, −9.947081104368732497996672958015, −8.988448964333001409784136164352, −8.228751937387595422601025583505, −7.08470808241160050925706982838, −6.58159352263424463045867180336, −5.48657669059509500897128805703, −4.18261287068864886038838632295, −2.44340959545908893695303794126, −0.854657942454831982931819724197, 1.52806388260460076721320882733, 3.11544074081659525400107140282, 4.25496581255278465012870330881, 5.45382355835451977281309901833, 6.55120882445338372889867370042, 7.74359973530997713774106734492, 8.679922436872612618420970964012, 9.561971928335584022913248310679, 10.11477277104225010454901985303, 11.09693812997453871160839026730

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