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Properties

Label 2-462-33.8-c1-0-3
Degree $2$
Conductor $462$
Sign $-0.493 - 0.869i$
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 + (−1.63 + 0.569i)3-s + (0.309 − 0.951i)4-s + (−0.732 + 1.00i)5-s + (0.988 − 1.42i)6-s + (0.951 + 0.309i)7-s + (0.309 + 0.951i)8-s + (2.35 − 1.86i)9-s − 1.24i·10-s + (3.01 − 1.37i)11-s + (0.0358 + 1.73i)12-s + (1.55 + 2.14i)13-s + (−0.951 + 0.309i)14-s + (0.624 − 2.06i)15-s + (−0.809 − 0.587i)16-s + (0.0657 + 0.0477i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.944 + 0.328i)3-s + (0.154 − 0.475i)4-s + (−0.327 + 0.450i)5-s + (0.403 − 0.580i)6-s + (0.359 + 0.116i)7-s + (0.109 + 0.336i)8-s + (0.783 − 0.620i)9-s − 0.394i·10-s + (0.910 − 0.413i)11-s + (0.0103 + 0.499i)12-s + (0.431 + 0.594i)13-s + (−0.254 + 0.0825i)14-s + (0.161 − 0.533i)15-s + (−0.202 − 0.146i)16-s + (0.0159 + 0.0115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.493 - 0.869i$
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: \((2,\ 462,\ (\ :1/2),\ -0.493 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.334938 + 0.575474i\)
\(L(\frac12)\) \(\approx\) \(0.334938 + 0.575474i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 0.569i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-3.01 + 1.37i)T \)
good5 \( 1 + (0.732 - 1.00i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.55 - 2.14i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.0657 - 0.0477i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.03 - 1.96i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.41iT - 23T^{2} \)
29 \( 1 + (-0.273 + 0.843i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.202 - 0.147i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.01 - 3.12i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.35 - 4.18i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.86iT - 43T^{2} \)
47 \( 1 + (6.48 - 2.10i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.70 - 2.33i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.184 + 0.0600i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.87 - 6.70i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.46T + 67T^{2} \)
71 \( 1 + (-7.04 + 9.69i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-15.7 - 5.10i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.39 + 7.42i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.69 + 2.68i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 12.0iT - 89T^{2} \)
97 \( 1 + (-10.4 + 7.61i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.29586264969559644001393199497, −10.56184635580758734330630455380, −9.545598439505016955158726010840, −8.732299020719901821335707828116, −7.61427479467719822966870294631, −6.56075696456112359658659789047, −6.03978906695242772371339642093, −4.73698790434866506773900423455, −3.65910915697551548512036804703, −1.46982511776161792503369253245, 0.61167766090904651229778064685, 2.02546418798777442649600357540, 3.98740494410226257670341275230, 4.87089884962055905277562527199, 6.25003706252636461850911513356, 7.04263690681861421826863706103, 8.166130731185251588367330266502, 8.864411425653796292804179215011, 10.11781074277055190442973240912, 10.82452689311483562366529759191

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