↑

Properties

Label 2-540-180.139-c0-0-0
Degree $2$
Conductor $540$
Sign $0.342 - 0.939i$
Analytic cond. $0.269495$
Root an. cond. $0.519129$
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 + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·28-s + (−0.5 − 0.866i)29-s + (0.499 − 0.866i)32-s + 0.999·35-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·28-s + (−0.5 − 0.866i)29-s + (0.499 − 0.866i)32-s + 0.999·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(0.269495\)
Root analytic conductor: \(0.519129\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167150523\)
\(L(\frac12)\) \(\approx\) \(1.167150523\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
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

−11.70225816721088247338415274469, −10.05384256778273470565226863028, −9.146436524116860180669508754909, −8.478473002340239398692614497072, −7.70815261552396702511844814822, −6.44199065028002407760298267795, −5.52900259488754075856356566529, −4.97919979098446550649991846714, −3.75325010947034271814718782800, −2.12142809824458251968121301207, 1.65891548502932067179509079584, 2.93819880043402743187568511838, 4.01063383425975005748374740452, 5.06249212530784345061425221832, 6.18358575565569073566325391242, 7.06821756023737041524021857298, 8.277525887460268212554830754048, 9.499566059348226330290444137282, 10.23325456326375212320291222532, 10.89664684612161506203859970544

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