Properties

Label 2-3e6-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.777·2-s − 1.39·4-s + 2.37·5-s − 2.50·7-s − 2.64·8-s + 1.84·10-s + 3.14·11-s + 1.33·13-s − 1.94·14-s + 0.736·16-s + 6.27·17-s + 8.06·19-s − 3.31·20-s + 2.44·22-s + 4.05·23-s + 0.647·25-s + 1.03·26-s + 3.48·28-s − 9.28·29-s + 2.83·31-s + 5.85·32-s + 4.87·34-s − 5.94·35-s + 5.53·37-s + 6.27·38-s − 6.27·40-s − 7.10·41-s + ⋯
L(s)  = 1  + 0.549·2-s − 0.697·4-s + 1.06·5-s − 0.945·7-s − 0.933·8-s + 0.584·10-s + 0.947·11-s + 0.370·13-s − 0.519·14-s + 0.184·16-s + 1.52·17-s + 1.85·19-s − 0.741·20-s + 0.520·22-s + 0.845·23-s + 0.129·25-s + 0.203·26-s + 0.659·28-s − 1.72·29-s + 0.508·31-s + 1.03·32-s + 0.836·34-s − 1.00·35-s + 0.909·37-s + 1.01·38-s − 0.992·40-s − 1.10·41-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: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.958524500\)
\(L(\frac12)\) \(\approx\) \(1.958524500\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.777T + 2T^{2} \)
5 \( 1 - 2.37T + 5T^{2} \)
7 \( 1 + 2.50T + 7T^{2} \)
11 \( 1 - 3.14T + 11T^{2} \)
13 \( 1 - 1.33T + 13T^{2} \)
17 \( 1 - 6.27T + 17T^{2} \)
19 \( 1 - 8.06T + 19T^{2} \)
23 \( 1 - 4.05T + 23T^{2} \)
29 \( 1 + 9.28T + 29T^{2} \)
31 \( 1 - 2.83T + 31T^{2} \)
37 \( 1 - 5.53T + 37T^{2} \)
41 \( 1 + 7.10T + 41T^{2} \)
43 \( 1 - 2.33T + 43T^{2} \)
47 \( 1 + 4.61T + 47T^{2} \)
53 \( 1 - 0.135T + 53T^{2} \)
59 \( 1 + 3.99T + 59T^{2} \)
61 \( 1 - 0.341T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 - 8.19T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 4.08T + 79T^{2} \)
83 \( 1 - 0.913T + 83T^{2} \)
89 \( 1 + 3.72T + 89T^{2} \)
97 \( 1 + 5.99T + 97T^{2} \)
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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

−9.944575102831909031611897207087, −9.617617743035539646279669451666, −9.058750539178337873642419260258, −7.75617541647280971092780424542, −6.57900993088727946801195254603, −5.76317037820004068980446124346, −5.19334468670348608286477884622, −3.72501173676307956100822876708, −3.09116245082537454398632213836, −1.20821513091691800461851737607, 1.20821513091691800461851737607, 3.09116245082537454398632213836, 3.72501173676307956100822876708, 5.19334468670348608286477884622, 5.76317037820004068980446124346, 6.57900993088727946801195254603, 7.75617541647280971092780424542, 9.058750539178337873642419260258, 9.617617743035539646279669451666, 9.944575102831909031611897207087

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