Properties

Label 2-462-77.54-c1-0-12
Degree $2$
Conductor $462$
Sign $0.875 - 0.483i$
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  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (2.83 + 1.63i)5-s + 0.999·6-s + (2.54 − 0.732i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1.63 + 2.83i)10-s + (−2.54 − 2.12i)11-s + (0.866 + 0.499i)12-s − 5.12·13-s + (2.56 + 0.637i)14-s + 3.26·15-s + (−0.5 + 0.866i)16-s + (−2.66 − 4.61i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (1.26 + 0.730i)5-s + 0.408·6-s + (0.960 − 0.276i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.516 + 0.895i)10-s + (−0.767 − 0.641i)11-s + (0.249 + 0.144i)12-s − 1.42·13-s + (0.686 + 0.170i)14-s + 0.843·15-s + (−0.125 + 0.216i)16-s + (−0.645 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.64421 + 0.681741i\)
\(L(\frac12)\) \(\approx\) \(2.64421 + 0.681741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-2.54 + 0.732i)T \)
11 \( 1 + (2.54 + 2.12i)T \)
good5 \( 1 + (-2.83 - 1.63i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 + (2.66 + 4.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.13 - 3.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.68 - 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.77iT - 29T^{2} \)
31 \( 1 + (-5.59 + 3.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.06 - 1.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.949T + 41T^{2} \)
43 \( 1 - 6.20iT - 43T^{2} \)
47 \( 1 + (-10.4 - 6.01i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.26 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.6 - 6.74i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.67 + 4.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.09 + 5.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + (-6.61 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.93 - 4.58i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.835T + 83T^{2} \)
89 \( 1 + (5.39 + 3.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.624iT - 97T^{2} \)
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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

−11.17349678633124619349890851608, −10.17800274023861838122272492371, −9.433153231796021991618495869557, −8.049696484853960528977550927730, −7.48799330659194680738683793326, −6.39889549869189592345473460477, −5.50745635888024154888342928278, −4.48709370127456046793441854541, −2.83667443900854372000207353040, −2.09420809459570619048205245717, 1.91007684713703292971761979037, 2.50443971542512371821989616311, 4.56680466873940324076452129820, 4.88581879840322262397502494565, 5.97052218286497219656174655313, 7.29397292882314341301314155622, 8.523300353991730157729734686741, 9.205385282798814558277371230864, 10.29815073743265087305612686207, 10.67058163888073818192157888996

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