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Properties

Label 2-462-11.5-c1-0-3
Degree $2$
Conductor $462$
Sign $-0.789 - 0.613i$
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 + (−1.02 + 3.14i)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 − 3.30·10-s + (3.14 − 1.06i)11-s − 0.999·12-s + (0.682 + 2.10i)13-s + (0.809 + 0.587i)14-s + (−2.67 + 1.94i)15-s + (0.309 − 0.951i)16-s + (−2.16 + 6.65i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.457 + 1.40i)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 − 1.04·10-s + (0.946 − 0.321i)11-s − 0.288·12-s + (0.189 + 0.582i)13-s + (0.216 + 0.157i)14-s + (−0.691 + 0.502i)15-s + (0.0772 − 0.237i)16-s + (−0.524 + 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.789 - 0.613i$
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: \((2,\ 462,\ (\ :1/2),\ -0.789 - 0.613i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515614 + 1.50369i\)
\(L(\frac12)\) \(\approx\) \(0.515614 + 1.50369i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.14 + 1.06i)T \)
good5 \( 1 + (1.02 - 3.14i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.682 - 2.10i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.16 - 6.65i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.44 + 3.95i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (-5.65 + 4.10i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.09 + 3.38i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.55 - 1.85i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.54 - 6.20i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (0.317 + 0.230i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.11 - 6.49i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.19 + 4.50i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.30 + 4.01i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.06T + 67T^{2} \)
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 \( 1 + (-2.25 - 6.92i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.02 - 6.22i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 3.70T + 89T^{2} \)
97 \( 1 + (0.223 + 0.688i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.11979820761243512955424587923, −10.70320691721017629962256628168, −9.488696083372936859402417410883, −8.495637148483995837281463478183, −7.77371592629136080641259419841, −6.58878130571766151186170286716, −6.24022618234713250243469490440, −4.26597441165724253736034635262, −3.88152642232444425854474697580, −2.39562133022619697385680992121, 0.928672317040207294263118471739, 2.26288416466104194594803284589, 3.86893981501854719585761385144, 4.62487732941697957194370941727, 5.73321449572142907865551177370, 7.14056804711631903335439509854, 8.358967522421980137023980059274, 8.813374764469434370602439301315, 9.645887694061333823373883660875, 10.83066412397636890992858856495

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