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Properties

Label 2-462-77.13-c1-0-5
Degree $2$
Conductor $462$
Sign $0.714 - 0.700i$
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  + (−0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (1.43 − 0.467i)5-s + (−0.309 − 0.951i)6-s + (2.01 + 1.71i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s − 1.51·10-s + (0.803 + 3.21i)11-s + 0.999i·12-s + (−1.11 + 3.43i)13-s + (−1.38 − 2.25i)14-s + (1.22 + 0.889i)15-s + (0.309 + 0.951i)16-s + (−2.43 − 7.50i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 + 0.293i)4-s + (0.643 − 0.209i)5-s + (−0.126 − 0.388i)6-s + (0.762 + 0.647i)7-s + (−0.207 − 0.286i)8-s + (−0.103 + 0.317i)9-s − 0.478·10-s + (0.242 + 0.970i)11-s + 0.288i·12-s + (−0.309 + 0.951i)13-s + (−0.371 − 0.601i)14-s + (0.315 + 0.229i)15-s + (0.0772 + 0.237i)16-s + (−0.591 − 1.82i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.714 - 0.700i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.714 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25318 + 0.511767i\)
\(L(\frac12)\) \(\approx\) \(1.25318 + 0.511767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (-2.01 - 1.71i)T \)
11 \( 1 + (-0.803 - 3.21i)T \)
good5 \( 1 + (-1.43 + 0.467i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.11 - 3.43i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.43 + 7.50i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.13 + 0.826i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.44T + 23T^{2} \)
29 \( 1 + (2.52 - 3.47i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.641 + 0.208i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-9.55 - 6.93i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.97 + 1.43i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.927iT - 43T^{2} \)
47 \( 1 + (1.75 + 2.40i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.341 + 1.05i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.24 + 8.60i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.413 + 1.27i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.63T + 67T^{2} \)
71 \( 1 + (1.38 + 4.26i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.69 + 4.13i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (14.5 + 4.72i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.33 - 10.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.35iT - 89T^{2} \)
97 \( 1 + (-7.04 - 2.28i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.32044284972479685957788140945, −9.900493337628052693937106848481, −9.351843556694208446466952522570, −8.867935913693514167416261245452, −7.60648293516593110010818763763, −6.78021611796898498689217456749, −5.26133479079876112264904727194, −4.50938461662491602074204191498, −2.73242276269609417461966657960, −1.76297904920216460092232375683, 1.12167223293253842943699628050, 2.44188926738381564238000761966, 3.95284920145053053121377755967, 5.62830159777568020714668248619, 6.32617360528471215933173794750, 7.52877479404886913641174503765, 8.152296444923814341695805784037, 8.979241743403888279542606653584, 10.08609709252923834776998135112, 10.79946585717669333661794326416

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