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Properties

Label 2-6e3-72.11-c1-0-8
Degree $2$
Conductor $216$
Sign $0.827 + 0.561i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 − 0.111i)2-s + (1.97 − 0.313i)4-s + (−1.74 − 3.01i)5-s + (1.80 + 1.04i)7-s + (2.75 − 0.660i)8-s + (−2.79 − 4.06i)10-s + (0.116 + 0.0675i)11-s + (−2.63 + 1.52i)13-s + (2.66 + 1.27i)14-s + (3.80 − 1.23i)16-s + 4.19i·17-s + 0.919·19-s + (−4.38 − 5.41i)20-s + (0.172 + 0.0822i)22-s + (−0.689 − 1.19i)23-s + ⋯
L(s)  = 1  + (0.996 − 0.0785i)2-s + (0.987 − 0.156i)4-s + (−0.779 − 1.35i)5-s + (0.683 + 0.394i)7-s + (0.972 − 0.233i)8-s + (−0.883 − 1.28i)10-s + (0.0352 + 0.0203i)11-s + (−0.731 + 0.422i)13-s + (0.712 + 0.339i)14-s + (0.950 − 0.309i)16-s + 1.01i·17-s + 0.210·19-s + (−0.981 − 1.21i)20-s + (0.0367 + 0.0175i)22-s + (−0.143 − 0.249i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92588 - 0.592120i\)
\(L(\frac12)\) \(\approx\) \(1.92588 - 0.592120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.111i)T \)
3 \( 1 \)
good5 \( 1 + (1.74 + 3.01i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.80 - 1.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.116 - 0.0675i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.63 - 1.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.19iT - 17T^{2} \)
19 \( 1 - 0.919T + 19T^{2} \)
23 \( 1 + (0.689 + 1.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.24 - 7.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.39 - 2.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.61iT - 37T^{2} \)
41 \( 1 + (1.79 - 1.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.41 + 9.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.205 + 0.356i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.968T + 53T^{2} \)
59 \( 1 + (3.88 - 2.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.44 + 4.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.15 - 5.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 4.06T + 73T^{2} \)
79 \( 1 + (-10.8 - 6.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.23 + 3.02i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.35iT - 89T^{2} \)
97 \( 1 + (0.477 - 0.826i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.39448332633315243426814859395, −11.62742740329206703096343988884, −10.65295709593479105764791267967, −9.118961229574228390294639429530, −8.160770902026965056753138793805, −7.14389364363933566126875219750, −5.54506883490018869721635641917, −4.78686368056344647691564152110, −3.77199638556114063993728877702, −1.77177590401366217256348352551, 2.54740120379529009673904208410, 3.70262532576811738699124413137, 4.86891182230800599810423251916, 6.24054866562548459538189749059, 7.52738941690980668254916070429, 7.65214447984692564665366876371, 9.799833319707830733609324694871, 10.98622387314225090154223809697, 11.37222208964884709525538849161, 12.30470657724991087696123241060

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