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Properties

Label 2-462-11.5-c1-0-10
Degree $2$
Conductor $462$
Sign $0.190 + 0.981i$
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 + (0.910 − 2.80i)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 + 2.94·10-s + (−3.06 − 1.26i)11-s + 0.999·12-s + (−0.164 − 0.504i)13-s + (−0.809 − 0.587i)14-s + (−2.38 + 1.73i)15-s + (0.309 − 0.951i)16-s + (2.26 − 6.97i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.407 − 1.25i)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 + 0.931·10-s + (−0.924 − 0.380i)11-s + 0.288·12-s + (−0.0455 − 0.140i)13-s + (−0.216 − 0.157i)14-s + (−0.615 + 0.447i)15-s + (0.0772 − 0.237i)16-s + (0.549 − 1.69i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.190 + 0.981i$
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: \((2,\ 462,\ (\ :1/2),\ 0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749045 - 0.617553i\)
\(L(\frac12)\) \(\approx\) \(0.749045 - 0.617553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.06 + 1.26i)T \)
good5 \( 1 + (-0.910 + 2.80i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.164 + 0.504i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.26 + 6.97i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.35 + 3.16i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.710T + 23T^{2} \)
29 \( 1 + (-0.0769 + 0.0559i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.55 + 4.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.85 + 1.34i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.88 - 1.37i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.172T + 43T^{2} \)
47 \( 1 + (-10.6 - 7.77i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.22 + 3.77i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.40 + 5.38i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.37 - 7.30i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + (-2.74 + 8.44i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.25 - 3.82i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.17 - 9.78i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.54 - 13.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.61T + 89T^{2} \)
97 \( 1 + (0.280 + 0.863i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.93884251218759988123711768245, −9.694975118471805973481641221511, −8.987805114547376672188621144472, −8.040910036141654047075913894107, −7.14680602935822659269283875441, −5.92925135024366969555566694436, −5.30881160132805279313160447099, −4.47284360437218767303202342343, −2.62722582157417563419684918311, −0.57834052017117307490216642339, 2.01545090936526859601004338116, 3.26605840146228804163506497286, 4.26020494526862486560584087304, 5.66250404789323429528723026501, 6.35052756812359158840872044669, 7.48368246822593471361243845136, 8.725585490811089585070683904409, 10.14055318886798847582824661502, 10.34978492435752208563821788360, 10.89610742331124448162299173595

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