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Properties

Label 2-462-11.5-c1-0-6
Degree $2$
Conductor $462$
Sign $0.502 - 0.864i$
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 + (0.640 − 1.97i)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 + 2.07·10-s + (1.47 + 2.96i)11-s − 0.999·12-s + (1.31 + 4.05i)13-s + (0.809 + 0.587i)14-s + (1.67 − 1.21i)15-s + (0.309 − 0.951i)16-s + (1.16 − 3.57i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.286 − 0.881i)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 + 0.655·10-s + (0.445 + 0.895i)11-s − 0.288·12-s + (0.365 + 1.12i)13-s + (0.216 + 0.157i)14-s + (0.432 − 0.314i)15-s + (0.0772 − 0.237i)16-s + (0.281 − 0.867i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68658 + 0.970561i\)
\(L(\frac12)\) \(\approx\) \(1.68658 + 0.970561i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 - 2.96i)T \)
good5 \( 1 + (-0.640 + 1.97i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.31 - 4.05i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.16 + 3.57i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.59 - 1.15i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.63T + 23T^{2} \)
29 \( 1 + (2.41 - 1.75i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.51 + 7.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.91 - 5.02i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.54 + 4.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.20T + 43T^{2} \)
47 \( 1 + (-0.317 - 0.230i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.87 + 8.85i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.50 + 2.54i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.359 - 1.10i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + (-3.85 + 11.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.69 - 4.13i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.27 - 10.0i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.96 + 9.12i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.94T + 89T^{2} \)
97 \( 1 + (-2.22 - 6.84i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.32541387632956259673990927713, −9.930938319575241339204805157463, −9.199651649164613886567200028788, −8.660833507206916630123410428007, −7.45164216508497360202638987337, −6.72856681570468514143480901176, −5.21301007067062077263983337891, −4.68467447657946187709975818834, −3.52448159077187375840119811180, −1.64805434068831383621277789572, 1.38924013915876499992568784993, 2.93558402047277417262529655753, 3.48083114142073694700765272961, 5.18539745851581414258967979057, 6.17082234136551419192578633784, 7.20474692398601820001193405870, 8.412730891532237428957963865406, 9.029731057063415343999017178239, 10.34832310904716755663347382411, 10.79778247626672044068376756214

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