Properties

Label 2-3e6-27.23-c0-0-0
Degree $2$
Conductor $729$
Sign $0.727 - 0.686i$
Analytic cond. $0.363818$
Root an. cond. $0.603173$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)7-s + (0.939 − 0.342i)13-s + (0.173 + 0.984i)16-s + (0.5 + 0.866i)19-s + (−0.939 − 0.342i)25-s − 28-s + (−0.766 − 0.642i)31-s + (0.5 − 0.866i)37-s + (−0.173 − 0.984i)43-s + (0.939 + 0.342i)52-s + (1.53 − 1.28i)61-s + (−0.500 + 0.866i)64-s + (−1.87 + 0.684i)67-s + (−1 − 1.73i)73-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)7-s + (0.939 − 0.342i)13-s + (0.173 + 0.984i)16-s + (0.5 + 0.866i)19-s + (−0.939 − 0.342i)25-s − 28-s + (−0.766 − 0.642i)31-s + (0.5 − 0.866i)37-s + (−0.173 − 0.984i)43-s + (0.939 + 0.342i)52-s + (1.53 − 1.28i)61-s + (−0.500 + 0.866i)64-s + (−1.87 + 0.684i)67-s + (−1 − 1.73i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(0.363818\)
Root analytic conductor: \(0.603173\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :0),\ 0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066492985\)
\(L(\frac12)\) \(\approx\) \(1.066492985\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
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

−10.78655664230508858058466287167, −9.865997572810064271723386070306, −8.931345010869466592611469397164, −8.067035090547322127561812305636, −7.28573264236949235459241875763, −6.17674542936378892659000986287, −5.70568743806275732140333888411, −3.94674040639275022643603232554, −3.18051263007431464899492891317, −1.99295865601455482815840469047, 1.33480327883852614411324441832, 2.83779643984027513884424658112, 3.91224601865939168016083475387, 5.24195622390657075321901953825, 6.23719818021986058979909876668, 6.86308725391844403851833280350, 7.70154228999914851549681228473, 8.971923348024065487993301955183, 9.774200664428923961296131134324, 10.48852580581089681161920607294

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