↑

Properties

Label 2-540-540.139-c0-0-1
Degree $2$
Conductor $540$
Sign $-0.448 + 0.893i$
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  + (−0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)5-s + (−0.5 + 0.866i)6-s + (0.266 − 0.223i)7-s + (−0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (−0.326 + 0.118i)14-s + (−0.939 − 0.342i)15-s + (0.173 + 0.984i)16-s + (0.766 + 0.642i)18-s + (0.766 − 0.642i)20-s + (−0.173 − 0.300i)21-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)5-s + (−0.5 + 0.866i)6-s + (0.266 − 0.223i)7-s + (−0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (−0.326 + 0.118i)14-s + (−0.939 − 0.342i)15-s + (0.173 + 0.984i)16-s + (0.766 + 0.642i)18-s + (0.766 − 0.642i)20-s + (−0.173 − 0.300i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $-0.448 + 0.893i$
Analytic conductor: \(0.269495\)
Root analytic conductor: \(0.519129\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :0),\ -0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6343862055\)
\(L(\frac12)\) \(\approx\) \(0.6343862055\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
good7 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 - 0.984i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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

−10.92395316676460273617763346531, −9.455980013000982829931151209037, −9.109718242720693799342522656728, −7.993423241124774073970108365598, −7.54934930684199580968423009673, −6.42038538853434351960634738268, −5.32242728259071826141955425749, −3.68700664288950873877250099158, −2.21921498987365006306752534309, −1.09992931266944441448079101524, 2.27170527121209624111704454607, 3.35602399795940583148537027411, 4.93342751371940624959610212884, 5.93208367847989290564859844565, 6.89456024139881863782814390411, 7.897009547647493388877120930688, 8.807533424339268777258134074595, 9.588035341398798064171088788370, 10.30391693998486811275889804233, 11.12604131928710898887668369125

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