↑

Properties

Label 2-2340-260.179-c0-0-1
Degree $2$
Conductor $2340$
Sign $-0.872 + 0.488i$
Analytic cond. $1.16781$
Root an. cond. $1.08065$
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 + 0.999·8-s − 0.999·10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (−1.5 − 0.866i)17-s + (0.499 + 0.866i)20-s + (−0.499 − 0.866i)25-s + (−0.499 + 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s + 1.73i·34-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s − 0.999·10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (−1.5 − 0.866i)17-s + (0.499 + 0.866i)20-s + (−0.499 − 0.866i)25-s + (−0.499 + 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s + 1.73i·34-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ -0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7522175054\)
\(L(\frac12)\) \(\approx\) \(0.7522175054\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.972337002872417002452170565700, −8.401778810997543046604790358185, −7.51601375782960703317785784678, −6.69019193514015482135868879667, −5.40167907940970704936161584671, −4.82929680043790461930194013427, −3.92012694808935278542603996351, −2.73191806492029992542916402400, −1.96000304757661680868619476886, −0.59399449254363931391372537060, 1.67726933105061520128119459838, 2.63513941741403956007444998458, 4.12843145676526470424961264557, 4.78968537723391802193478270294, 6.05369756595507505462621147984, 6.36594065584998593666017584046, 7.11874892192957121085948739382, 7.87082337847413851745786429646, 8.738941703944119702979204921167, 9.451415212167278172449171789783

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