↑

Properties

Label 2-720-144.11-c1-0-23
Degree $2$
Conductor $720$
Sign $0.101 - 0.994i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 + 0.204i)2-s + (−1.63 + 0.568i)3-s + (1.91 − 0.573i)4-s + (0.965 + 0.258i)5-s + (2.17 − 1.13i)6-s + (−0.548 + 0.950i)7-s + (−2.56 + 1.19i)8-s + (2.35 − 1.86i)9-s + (−1.40 − 0.164i)10-s + (−1.50 + 0.402i)11-s + (−2.80 + 2.02i)12-s + (4.22 + 1.13i)13-s + (0.573 − 1.44i)14-s + (−1.72 + 0.126i)15-s + (3.34 − 2.19i)16-s + 1.47i·17-s + ⋯
L(s)  = 1  + (−0.989 + 0.144i)2-s + (−0.944 + 0.328i)3-s + (0.958 − 0.286i)4-s + (0.431 + 0.115i)5-s + (0.886 − 0.461i)6-s + (−0.207 + 0.359i)7-s + (−0.906 + 0.422i)8-s + (0.784 − 0.620i)9-s + (−0.444 − 0.0519i)10-s + (−0.452 + 0.121i)11-s + (−0.810 + 0.585i)12-s + (1.17 + 0.314i)13-s + (0.153 − 0.385i)14-s + (−0.446 + 0.0325i)15-s + (0.835 − 0.549i)16-s + 0.357i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.101 - 0.994i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.508885 + 0.459477i\)
\(L(\frac12)\) \(\approx\) \(0.508885 + 0.459477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.204i)T \)
3 \( 1 + (1.63 - 0.568i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 + (0.548 - 0.950i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.50 - 0.402i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-4.22 - 1.13i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 1.47iT - 17T^{2} \)
19 \( 1 + (1.67 + 1.67i)T + 19iT^{2} \)
23 \( 1 + (0.617 - 0.356i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.39 + 0.643i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-5.84 + 3.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.856 - 0.856i)T + 37iT^{2} \)
41 \( 1 + (0.707 + 1.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.42 - 5.30i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.74 + 3.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.14 - 9.14i)T - 53iT^{2} \)
59 \( 1 + (2.39 - 8.95i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.05 - 7.67i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.40 - 5.25i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 5.30iT - 73T^{2} \)
79 \( 1 + (-5.93 - 3.42i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.01 - 11.2i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 1.67T + 89T^{2} \)
97 \( 1 + (-5.79 + 10.0i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.55823772489893024389791670095, −9.820577494687364807882177292682, −9.042188587485248275589757999043, −8.162328928286955836776237480662, −6.98034753598618677559357690716, −6.18000817429535997325151579579, −5.66505368353394263257497825218, −4.28274569832072460256765005018, −2.70606863810217394642341414592, −1.20600807199205797977017299885, 0.64174862505454314221014272815, 1.89987340549761988949159973634, 3.41540811658470501412522947210, 4.96432955880439530242452829417, 6.13692417342212788944262150952, 6.57773852585435470712318706723, 7.70534659682304259997152873781, 8.423561500587841059560526876393, 9.510530125517678987035342424050, 10.42841063496736585792917333113

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