Properties

Label 2-3e6-1.1-c1-0-29
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  + 1.96·2-s + 1.87·4-s − 3.70·5-s − 2.34·7-s − 0.237·8-s − 7.29·10-s + 2.17·11-s − 4.71·13-s − 4.62·14-s − 4.22·16-s + 2.93·17-s − 6.22·19-s − 6.95·20-s + 4.29·22-s − 0.519·23-s + 8.70·25-s − 9.29·26-s − 4.41·28-s + 3.49·29-s − 4.30·31-s − 7.84·32-s + 5.78·34-s + 8.68·35-s + 2.41·37-s − 12.2·38-s + 0.879·40-s − 2.49·41-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.939·4-s − 1.65·5-s − 0.887·7-s − 0.0839·8-s − 2.30·10-s + 0.656·11-s − 1.30·13-s − 1.23·14-s − 1.05·16-s + 0.711·17-s − 1.42·19-s − 1.55·20-s + 0.914·22-s − 0.108·23-s + 1.74·25-s − 1.82·26-s − 0.833·28-s + 0.648·29-s − 0.773·31-s − 1.38·32-s + 0.991·34-s + 1.46·35-s + 0.396·37-s − 1.98·38-s + 0.139·40-s − 0.390·41-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 \( 1 - 1.96T + 2T^{2} \)
5 \( 1 + 3.70T + 5T^{2} \)
7 \( 1 + 2.34T + 7T^{2} \)
11 \( 1 - 2.17T + 11T^{2} \)
13 \( 1 + 4.71T + 13T^{2} \)
17 \( 1 - 2.93T + 17T^{2} \)
19 \( 1 + 6.22T + 19T^{2} \)
23 \( 1 + 0.519T + 23T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 + 4.30T + 31T^{2} \)
37 \( 1 - 2.41T + 37T^{2} \)
41 \( 1 + 2.49T + 41T^{2} \)
43 \( 1 - 1.06T + 43T^{2} \)
47 \( 1 - 0.237T + 47T^{2} \)
53 \( 1 + 4.66T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 3.67T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + 1.20T + 71T^{2} \)
73 \( 1 + 4.68T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + 0.699T + 89T^{2} \)
97 \( 1 + 7.08T + 97T^{2} \)
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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

−10.10313640309687283240335329892, −9.063424773050280171075649177809, −8.050705876264435288243875441721, −7.04808265347112877027198488002, −6.42692460015525069859320627369, −5.17535371438318617054043815800, −4.23515360459893042182284278796, −3.67069318112554478018041202332, −2.68660868794373027615249931288, 0, 2.68660868794373027615249931288, 3.67069318112554478018041202332, 4.23515360459893042182284278796, 5.17535371438318617054043815800, 6.42692460015525069859320627369, 7.04808265347112877027198488002, 8.050705876264435288243875441721, 9.063424773050280171075649177809, 10.10313640309687283240335329892

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