↑

Properties

Label 2-684-684.463-c0-0-0
Degree $2$
Conductor $684$
Sign $0.466 + 0.884i$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s − 5-s + (0.866 − 0.499i)6-s + (0.866 − 0.5i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s − 0.999i·12-s − 0.999i·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s − 5-s + (0.866 − 0.499i)6-s + (0.866 − 0.5i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s − 0.999i·12-s − 0.999i·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 0.999·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.466 + 0.884i$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ 0.466 + 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373993975\)
\(L(\frac12)\) \(\approx\) \(1.373993975\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 - iT \)
good5 \( 1 + T + T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.85025469826238630187836194214, −9.675845269837978080642935153950, −9.001217327796155657766829514712, −8.070021342105529722519544891961, −7.29951906259374418992540979270, −5.73478425319358405516873102911, −4.43417628710055703206611472687, −4.05470540489822477627501766767, −3.07735947729980339037787847107, −1.61416193566739030972483670363, 2.09211131883567116904745580854, 3.62681364608755873518369191631, 4.23372977425948726348259269171, 5.41677370785096554987554671439, 6.77373344023261663939376251516, 7.26125360817342554733913492351, 8.246017679071168183473204340461, 8.666408305888561454748719129582, 9.508027313020152184984742739873, 11.19916686201805429887553472948

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