↑

Properties

Label 2-3240-360.139-c0-0-0
Degree $2$
Conductor $3240$
Sign $-0.173 - 0.984i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
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 + (1 + 1.73i)11-s + (−0.5 + 0.866i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 19-s + (−0.499 − 0.866i)20-s + (0.999 − 1.73i)22-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·26-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 + (1 + 1.73i)11-s + (−0.5 + 0.866i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 19-s + (−0.499 − 0.866i)20-s + (0.999 − 1.73i)22-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (2539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4095321184\)
\(L(\frac12)\) \(\approx\) \(0.4095321184\)
\(L(1)\) 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 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
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

−9.250648848181753343840710991336, −8.422186182482200567766248301543, −7.40954608538076134563481673061, −6.94895603657755259193624640769, −6.54809510236037711097521435193, −4.64139054334581828293505593926, −4.27755478882460657635391282139, −3.50679612418110058989646829639, −2.47154083436437042987522116496, −1.58439722650982509814668325634, 0.29806711929197286982440614761, 1.59632432546985261808095667094, 3.18867958611158477146406243618, 3.99523513629948252906578176199, 5.19779706484269263561797165010, 5.59159254068738686641822853341, 6.35755515385569860338094178747, 7.15489562445487458480781604003, 8.122059813772090837011813955465, 8.677963390654611546023830711172

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