↑

Properties

Label 2-720-20.19-c2-0-27
Degree $2$
Conductor $720$
Sign $0.5 + 0.866i$
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  + 5·5-s + 6.92·7-s − 13.8i·11-s − 24i·13-s + 24i·17-s − 27.7i·19-s − 34.6·23-s + 25·25-s − 10·29-s − 13.8i·31-s + 34.6·35-s + 24i·37-s + 34·41-s + 20.7·43-s − 6.92·47-s + ⋯
L(s)  = 1  + 5-s + 0.989·7-s − 1.25i·11-s − 1.84i·13-s + 1.41i·17-s − 1.45i·19-s − 1.50·23-s + 25-s − 0.344·29-s − 0.446i·31-s + 0.989·35-s + 0.648i·37-s + 0.829·41-s + 0.483·43-s − 0.147·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.5 + 0.866i$
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: \((2,\ 720,\ (\ :1),\ 0.5 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.327904584\)
\(L(\frac12)\) \(\approx\) \(2.327904584\)
\(L(2)\) 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 \( 1 - 5T \)
good7 \( 1 - 6.92T + 49T^{2} \)
11 \( 1 + 13.8iT - 121T^{2} \)
13 \( 1 + 24iT - 169T^{2} \)
17 \( 1 - 24iT - 289T^{2} \)
19 \( 1 + 27.7iT - 361T^{2} \)
23 \( 1 + 34.6T + 529T^{2} \)
29 \( 1 + 10T + 841T^{2} \)
31 \( 1 + 13.8iT - 961T^{2} \)
37 \( 1 - 24iT - 1.36e3T^{2} \)
41 \( 1 - 34T + 1.68e3T^{2} \)
43 \( 1 - 20.7T + 1.84e3T^{2} \)
47 \( 1 + 6.92T + 2.20e3T^{2} \)
53 \( 1 - 48iT - 2.80e3T^{2} \)
59 \( 1 + 13.8iT - 3.48e3T^{2} \)
61 \( 1 - 70T + 3.72e3T^{2} \)
67 \( 1 + 90.0T + 4.48e3T^{2} \)
71 \( 1 + 55.4iT - 5.04e3T^{2} \)
73 \( 1 + 48iT - 5.32e3T^{2} \)
79 \( 1 + 41.5iT - 6.24e3T^{2} \)
83 \( 1 - 90.0T + 6.88e3T^{2} \)
89 \( 1 - 14T + 7.92e3T^{2} \)
97 \( 1 - 96iT - 9.40e3T^{2} \)
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

−10.31527281922936358209801944700, −9.122213539388492099825270098590, −8.305485871304281729564396758096, −7.71287702929072176432967726138, −6.15654455347882590867478682176, −5.75341672568722120469444311567, −4.73782485484767712413829166694, −3.34016834057295409023454527215, −2.18328405311313213166555257278, −0.835694250791680051347260399380, 1.66023917571681579179405076384, 2.20279777068237032636777196745, 4.09714730759728576549701523702, 4.86859941825498010978045105326, 5.86112292722128189501304316846, 6.89199608667027130683765194133, 7.64588040757718390205406262999, 8.788186468099153119674933020469, 9.623753923130955082304654617270, 10.09484095183783481428174532734

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