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Properties

Label 2-3e6-27.14-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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)7-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)16-s + (0.5 + 0.866i)19-s + (0.173 + 0.984i)25-s − 28-s + (0.939 − 0.342i)31-s + (0.5 − 0.866i)37-s + (−0.766 + 0.642i)43-s + (−0.173 − 0.984i)52-s + (−1.87 − 0.684i)61-s + (−0.500 + 0.866i)64-s + (0.347 − 1.96i)67-s + (−1 − 1.73i)73-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)7-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)16-s + (0.5 + 0.866i)19-s + (0.173 + 0.984i)25-s − 28-s + (0.939 − 0.342i)31-s + (0.5 − 0.866i)37-s + (−0.766 + 0.642i)43-s + (−0.173 − 0.984i)52-s + (−1.87 − 0.684i)61-s + (−0.500 + 0.866i)64-s + (0.347 − 1.96i)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: $\chi_{729} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :0),\ 0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8493336273\)
\(L(\frac12)\) \(\approx\) \(0.8493336273\)
\(L(1)\) 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 \( 1 + (0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.173 + 0.984i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
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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

−10.71343475221318595804443231018, −9.592040930190111692214303436407, −9.073948767431028297492851602817, −8.101609764971049034820272105992, −7.53903975070523437509804967260, −6.18459059752901128062402174739, −5.09415604745961991801379207341, −4.43165924689846831773873829699, −3.30006905457313040992511890378, −1.69316801652212957212058389030, 1.11039595924581354552842731452, 2.89788122762755178437030296667, 4.31598678520314317107371364768, 4.94697130196106096515591567895, 5.85019789089647864158040672938, 7.11394626090521797256455779443, 8.155167486862324897178058857488, 8.617491104520097752111525305875, 9.790311770213936977107662274657, 10.34579652374747927590146967109

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