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Properties

Label 2-462-33.8-c1-0-7
Degree $2$
Conductor $462$
Sign $0.0462 - 0.998i$
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 + (1.63 + 0.580i)3-s + (0.309 − 0.951i)4-s + (−0.571 + 0.786i)5-s + (−1.66 + 0.490i)6-s + (0.951 + 0.309i)7-s + (0.309 + 0.951i)8-s + (2.32 + 1.89i)9-s − 0.972i·10-s + (−2.84 + 1.70i)11-s + (1.05 − 1.37i)12-s + (1.75 + 2.40i)13-s + (−0.951 + 0.309i)14-s + (−1.38 + 0.952i)15-s + (−0.809 − 0.587i)16-s + (−1.52 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.942 + 0.334i)3-s + (0.154 − 0.475i)4-s + (−0.255 + 0.351i)5-s + (−0.678 + 0.200i)6-s + (0.359 + 0.116i)7-s + (0.109 + 0.336i)8-s + (0.775 + 0.631i)9-s − 0.307i·10-s + (−0.857 + 0.513i)11-s + (0.304 − 0.396i)12-s + (0.485 + 0.668i)13-s + (−0.254 + 0.0825i)14-s + (−0.358 + 0.245i)15-s + (−0.202 − 0.146i)16-s + (−0.369 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0462 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0462 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0462 - 0.998i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.0462 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01817 + 0.972100i\)
\(L(\frac12)\) \(\approx\) \(1.01817 + 0.972100i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 0.580i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (2.84 - 1.70i)T \)
good5 \( 1 + (0.571 - 0.786i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.75 - 2.40i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.52 + 1.10i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.57 + 1.48i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.74iT - 23T^{2} \)
29 \( 1 + (-1.68 + 5.18i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.90 - 1.38i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.449 + 1.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.867 + 2.67i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.83iT - 43T^{2} \)
47 \( 1 + (3.10 - 1.00i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.54 - 6.26i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-5.00 - 1.62i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.75 - 6.55i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (-8.37 + 11.5i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (9.90 + 3.21i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.154 + 0.213i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.173 - 0.126i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (-11.5 + 8.42i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.03641017025016893789249588926, −10.17738113176884332646807809907, −9.309577069262980352083691746231, −8.651068197067635208915340240036, −7.49007553608687492728226819836, −7.23808664335397703850865167083, −5.59646996117790238898123701788, −4.51030820548299052296916698494, −3.17843258834336995152153387291, −1.86613478094662383765177278689, 1.03168867031330781272948784292, 2.56803151496850301903966436629, 3.56246199458474848646395157102, 4.87753722351106821051422652239, 6.40232980955427651151686294763, 7.62606348946505041668934576507, 8.254051233768517911939215846404, 8.781387990420337282133961295902, 9.953331667408619549322318366247, 10.66698003995847241221941586659

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