↑

Properties

Label 2-6e3-72.61-c1-0-2
Degree $2$
Conductor $216$
Sign $-0.882 - 0.470i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
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  + (0.722 + 1.21i)2-s + (−0.956 + 1.75i)4-s + (−3.17 + 1.83i)5-s + (−0.191 + 0.332i)7-s + (−2.82 + 0.104i)8-s + (−4.51 − 2.53i)10-s + (1.73 + 1.00i)11-s + (−0.397 + 0.229i)13-s + (−0.542 + 0.00670i)14-s + (−2.16 − 3.36i)16-s + 4.08·17-s + 4.72i·19-s + (−0.180 − 7.32i)20-s + (0.0350 + 2.83i)22-s + (2.97 + 5.15i)23-s + ⋯
L(s)  = 1  + (0.510 + 0.859i)2-s + (−0.478 + 0.878i)4-s + (−1.41 + 0.819i)5-s + (−0.0725 + 0.125i)7-s + (−0.999 + 0.0370i)8-s + (−1.42 − 0.801i)10-s + (0.524 + 0.302i)11-s + (−0.110 + 0.0636i)13-s + (−0.145 + 0.00179i)14-s + (−0.542 − 0.840i)16-s + 0.990·17-s + 1.08i·19-s + (−0.0404 − 1.63i)20-s + (0.00747 + 0.605i)22-s + (0.620 + 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.882 - 0.470i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258287 + 1.03263i\)
\(L(\frac12)\) \(\approx\) \(0.258287 + 1.03263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.722 - 1.21i)T \)
3 \( 1 \)
good5 \( 1 + (3.17 - 1.83i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.191 - 0.332i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.73 - 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.397 - 0.229i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.08T + 17T^{2} \)
19 \( 1 - 4.72iT - 19T^{2} \)
23 \( 1 + (-2.97 - 5.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.03 - 1.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.592 - 1.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.74iT - 37T^{2} \)
41 \( 1 + (4.75 + 8.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.03 - 0.598i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.27 - 5.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.63iT - 53T^{2} \)
59 \( 1 + (0.603 - 0.348i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.23 - 2.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.87 + 5.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 + 2.68T + 73T^{2} \)
79 \( 1 + (5.35 - 9.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.49 - 3.16i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.56T + 89T^{2} \)
97 \( 1 + (2.98 - 5.17i)T + (-48.5 - 84.0i)T^{2} \)
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

−12.53371087487196494876479269646, −12.02022857669477620079894929623, −11.06440778397395377057653453146, −9.674106716531438781192412340048, −8.360839999339717722024620483076, −7.52688964187537224069707467790, −6.82486972954173720631971502641, −5.50401250174264233455219971891, −4.04873142824188897777592126614, −3.26835906253130334969557043092, 0.822201316537317741535202261830, 3.12938621361286864294156975158, 4.25374623731978715086068674330, 5.11255902642323115867903792745, 6.71400322847311509743681514680, 8.163812724259387902893434270101, 8.970741540987836390274086613592, 10.15156355088179090804545276453, 11.33172365982092061893809409035, 11.86932904425292255560453644048

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