Properties

Label 2-3e6-1.1-c1-0-13
Degree $2$
Conductor $729$
Sign $-1$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.11·2-s + 2.48·4-s − 2.68·5-s + 0.972·7-s − 1.01·8-s + 5.67·10-s − 0.316·11-s − 1.51·13-s − 2.05·14-s − 2.80·16-s + 1.17·17-s + 6.22·19-s − 6.65·20-s + 0.670·22-s + 2.16·23-s + 2.19·25-s + 3.20·26-s + 2.41·28-s − 4.40·29-s − 8.67·31-s + 7.97·32-s − 2.48·34-s − 2.60·35-s − 4.46·37-s − 13.1·38-s + 2.72·40-s + 5.84·41-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.24·4-s − 1.19·5-s + 0.367·7-s − 0.359·8-s + 1.79·10-s − 0.0955·11-s − 0.419·13-s − 0.550·14-s − 0.702·16-s + 0.284·17-s + 1.42·19-s − 1.48·20-s + 0.143·22-s + 0.450·23-s + 0.439·25-s + 0.628·26-s + 0.455·28-s − 0.817·29-s − 1.55·31-s + 1.41·32-s − 0.426·34-s − 0.440·35-s − 0.734·37-s − 2.13·38-s + 0.431·40-s + 0.912·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-1$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 2.11T + 2T^{2} \)
5 \( 1 + 2.68T + 5T^{2} \)
7 \( 1 - 0.972T + 7T^{2} \)
11 \( 1 + 0.316T + 11T^{2} \)
13 \( 1 + 1.51T + 13T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 - 6.22T + 19T^{2} \)
23 \( 1 - 2.16T + 23T^{2} \)
29 \( 1 + 4.40T + 29T^{2} \)
31 \( 1 + 8.67T + 31T^{2} \)
37 \( 1 + 4.46T + 37T^{2} \)
41 \( 1 - 5.84T + 41T^{2} \)
43 \( 1 - 5.59T + 43T^{2} \)
47 \( 1 + 2.47T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 1.72T + 59T^{2} \)
61 \( 1 - 1.01T + 61T^{2} \)
67 \( 1 - 0.856T + 67T^{2} \)
71 \( 1 + 9.59T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 5.54T + 97T^{2} \)
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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

−9.739283871366419286669688923613, −9.139227218070714036981462880339, −8.190764938875812809500694971431, −7.51735644548264790221693832423, −7.14388856231166287824695764269, −5.53051695146413229094843262166, −4.35960612079513948442785350673, −3.11748094666312729388666379753, −1.49984527792155789246925648260, 0, 1.49984527792155789246925648260, 3.11748094666312729388666379753, 4.35960612079513948442785350673, 5.53051695146413229094843262166, 7.14388856231166287824695764269, 7.51735644548264790221693832423, 8.190764938875812809500694971431, 9.139227218070714036981462880339, 9.739283871366419286669688923613

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