↑

Properties

Label 2-720-144.11-c1-0-17
Degree $2$
Conductor $720$
Sign $0.809 - 0.587i$
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  + (−1.41 − 0.0306i)2-s + (−0.425 − 1.67i)3-s + (1.99 + 0.0867i)4-s + (−0.965 − 0.258i)5-s + (0.549 + 2.38i)6-s + (−0.547 + 0.947i)7-s + (−2.82 − 0.183i)8-s + (−2.63 + 1.42i)9-s + (1.35 + 0.395i)10-s + (1.34 − 0.359i)11-s + (−0.703 − 3.39i)12-s + (−1.21 − 0.324i)13-s + (0.802 − 1.32i)14-s + (−0.0239 + 1.73i)15-s + (3.98 + 0.346i)16-s + 2.47i·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0216i)2-s + (−0.245 − 0.969i)3-s + (0.999 + 0.0433i)4-s + (−0.431 − 0.115i)5-s + (0.224 + 0.974i)6-s + (−0.206 + 0.358i)7-s + (−0.997 − 0.0650i)8-s + (−0.879 + 0.475i)9-s + (0.429 + 0.125i)10-s + (0.404 − 0.108i)11-s + (−0.203 − 0.979i)12-s + (−0.335 − 0.0899i)13-s + (0.214 − 0.353i)14-s + (−0.00617 + 0.447i)15-s + (0.996 + 0.0866i)16-s + 0.599i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.809 - 0.587i$
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: \((2,\ 720,\ (\ :1/2),\ 0.809 - 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550469 + 0.178588i\)
\(L(\frac12)\) \(\approx\) \(0.550469 + 0.178588i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0306i)T \)
3 \( 1 + (0.425 + 1.67i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (0.547 - 0.947i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.34 + 0.359i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.21 + 0.324i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 2.47iT - 17T^{2} \)
19 \( 1 + (-4.00 - 4.00i)T + 19iT^{2} \)
23 \( 1 + (7.36 - 4.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.269 - 0.0722i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-3.49 + 2.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.93 - 1.93i)T + 37iT^{2} \)
41 \( 1 + (1.99 + 3.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.34 - 8.73i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.24 + 5.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.39 + 9.39i)T - 53iT^{2} \)
59 \( 1 + (3.15 - 11.7i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.106 - 0.399i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.259 - 0.968i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + 2.72iT - 73T^{2} \)
79 \( 1 + (-1.00 - 0.582i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.72 - 10.1i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + (2.05 - 3.56i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.39534686354663300909971414644, −9.613509677312851157728606798537, −8.575461055522462966776239661781, −7.912431841844192259993107331683, −7.26742078323499247021870053958, −6.20778223629481014327498796382, −5.59078363362666109157002034586, −3.70216682463558766799536980773, −2.39983506518686931420016834034, −1.18963147357112825121731974547, 0.48205238590320985406662490991, 2.58359170146794161214230245060, 3.70469631671885833771387713044, 4.81749557770151917903402144920, 6.04112038749415105108368064455, 6.94751770644629332821650269020, 7.82992762009112941725775895503, 8.854259562175637183170116684025, 9.502457944107947838871797433001, 10.25762387045620975288253746769

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