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Properties

Label 2-462-77.6-c1-0-10
Degree $2$
Conductor $462$
Sign $0.0271 + 0.999i$
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 + (−0.0789 − 0.0256i)5-s + (−0.309 + 0.951i)6-s + (−1.18 − 2.36i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + 0.0830·10-s + (2.36 + 2.32i)11-s − 0.999i·12-s + (−0.764 − 2.35i)13-s + (1.86 + 1.88i)14-s + (−0.0672 + 0.0488i)15-s + (0.309 − 0.951i)16-s + (1.92 − 5.92i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.339 − 0.467i)3-s + (0.404 − 0.293i)4-s + (−0.0353 − 0.0114i)5-s + (−0.126 + 0.388i)6-s + (−0.449 − 0.893i)7-s + (−0.207 + 0.286i)8-s + (−0.103 − 0.317i)9-s + 0.0262·10-s + (0.714 + 0.699i)11-s − 0.288i·12-s + (−0.211 − 0.652i)13-s + (0.497 + 0.502i)14-s + (−0.0173 + 0.0126i)15-s + (0.0772 − 0.237i)16-s + (0.466 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0271 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0271 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0271 + 0.999i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.0271 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693715 - 0.675116i\)
\(L(\frac12)\) \(\approx\) \(0.693715 - 0.675116i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 2.36i)T \)
11 \( 1 + (-2.36 - 2.32i)T \)
good5 \( 1 + (0.0789 + 0.0256i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.764 + 2.35i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.92 + 5.92i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.47 + 2.52i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.15T + 23T^{2} \)
29 \( 1 + (-4.17 - 5.74i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-5.72 + 1.86i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.56 - 2.59i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.41 + 6.84i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.35iT - 43T^{2} \)
47 \( 1 + (-7.52 + 10.3i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.26 - 3.88i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.91 - 8.13i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.85 + 5.71i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (1.82 - 5.63i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.23 + 2.34i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-10.7 + 3.48i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.02 - 12.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 + (16.1 - 5.23i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.47884077250085276963522709822, −9.954433473071781522176109183512, −8.990446312122830288465038696136, −8.080594545292676336015997644071, −7.01536506556836277432319282116, −6.75480480126571147367875621519, −5.19216623155557637992474865304, −3.79143156542467992224610802253, −2.39730350362816510451036066610, −0.72172804135098924367809206046, 1.84588665025238617426900286066, 3.19926076190226932563372954510, 4.21898897390805265230367270799, 5.88042822277876559007118299352, 6.53911935976298856311250052386, 8.146183534530714716434417857549, 8.529999615647456426866888726763, 9.567498198333164290849945042456, 10.10358280075684882622998253625, 11.22082987260966949969317099434

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