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Properties

Label 2-462-33.29-c1-0-8
Degree $2$
Conductor $462$
Sign $-0.555 - 0.831i$
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 + (0.407 + 1.68i)3-s + (0.309 + 0.951i)4-s + (1.88 + 2.60i)5-s + (−0.660 + 1.60i)6-s + (−0.951 + 0.309i)7-s + (−0.309 + 0.951i)8-s + (−2.66 + 1.37i)9-s + 3.21i·10-s + (−0.355 − 3.29i)11-s + (−1.47 + 0.907i)12-s + (1.19 − 1.64i)13-s + (−0.951 − 0.309i)14-s + (−3.60 + 4.23i)15-s + (−0.809 + 0.587i)16-s + (1.57 − 1.14i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.235 + 0.971i)3-s + (0.154 + 0.475i)4-s + (0.844 + 1.16i)5-s + (−0.269 + 0.653i)6-s + (−0.359 + 0.116i)7-s + (−0.109 + 0.336i)8-s + (−0.889 + 0.457i)9-s + 1.01i·10-s + (−0.107 − 0.994i)11-s + (−0.425 + 0.261i)12-s + (0.332 − 0.457i)13-s + (−0.254 − 0.0825i)14-s + (−0.931 + 1.09i)15-s + (−0.202 + 0.146i)16-s + (0.382 − 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03524 + 1.93785i\)
\(L(\frac12)\) \(\approx\) \(1.03524 + 1.93785i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.407 - 1.68i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.355 + 3.29i)T \)
good5 \( 1 + (-1.88 - 2.60i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.19 + 1.64i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.57 + 1.14i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.14 - 0.698i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.70iT - 23T^{2} \)
29 \( 1 + (0.422 + 1.29i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.07 - 0.779i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.29 - 10.1i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.23 + 3.80i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.48iT - 43T^{2} \)
47 \( 1 + (-3.88 - 1.26i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.85 - 5.30i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (11.7 - 3.82i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.27 - 3.13i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.77T + 67T^{2} \)
71 \( 1 + (3.19 + 4.39i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.22 + 1.37i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.0 + 13.8i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.6 + 9.93i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 3.34iT - 89T^{2} \)
97 \( 1 + (8.66 + 6.29i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.08590146129007260034819029965, −10.50860902818169206019980110711, −9.703694589579710347112735139929, −8.687817039166584674877724645537, −7.68073222001729494705356794612, −6.27050741872623410447338373093, −5.91628232322051642306248324218, −4.67435708269497438278451734296, −3.24296245186967980196984478033, −2.78626080172514311630425510981, 1.25703663136750720507309054172, 2.21222570774238423367377442702, 3.74863152845225783671917953106, 5.13030281886291844771975369958, 5.86865730541815694155853452283, 6.94182016061207826744826824601, 7.942240749377303564927042598665, 9.284729347037659511220171593271, 9.528340294655300862330686343466, 10.92146676669771420444484043292

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