↑

Properties

Label 2-720-20.19-c2-0-20
Degree $2$
Conductor $720$
Sign $0.799 + 0.599i$
Analytic cond. $19.6185$
Root an. cond. $4.42928$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 3i)5-s − 24i·13-s − 30i·17-s + (7 + 24i)25-s + 40·29-s + 24i·37-s + 80·41-s − 49·49-s − 90i·53-s + 22·61-s + (72 − 96i)65-s + 96i·73-s + (90 − 120i)85-s + 160·89-s − 144i·97-s + ⋯
L(s)  = 1  + (0.800 + 0.600i)5-s − 1.84i·13-s − 1.76i·17-s + (0.280 + 0.959i)25-s + 1.37·29-s + 0.648i·37-s + 1.95·41-s − 0.999·49-s − 1.69i·53-s + 0.360·61-s + (1.10 − 1.47i)65-s + 1.31i·73-s + (1.05 − 1.41i)85-s + 1.79·89-s − 1.48i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.799 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.799 + 0.599i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ 0.799 + 0.599i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.062605578\)
\(L(\frac12)\) \(\approx\) \(2.062605578\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-4 - 3i)T \)
good7 \( 1 + 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 24iT - 169T^{2} \)
17 \( 1 + 30iT - 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 40T + 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 - 24iT - 1.36e3T^{2} \)
41 \( 1 - 80T + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 90iT - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 22T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 96iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 160T + 7.92e3T^{2} \)
97 \( 1 + 144iT - 9.40e3T^{2} \)
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

−10.06209609593778240058757712816, −9.509845417310531378175017399470, −8.352699868080623691552539794036, −7.46890873056342618583782140776, −6.57926182402594870502176612674, −5.61980897062721922562137901515, −4.85989479768828035598562976539, −3.20685489441831752310022918616, −2.53831365851164639560552336411, −0.797299987566476688261041837917, 1.33381644013624857409242843241, 2.31722429720029262513114357722, 3.99152190107949918441622133530, 4.74682511606462404615535684663, 6.02747037150496146758744486226, 6.50604283296682602516088397348, 7.79384236081318026139011622890, 8.807950454077778297592798289202, 9.291737438529966952632907849883, 10.25897143868498582204806844618

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