Properties

Label 2-3e6-1.1-c1-0-12
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  + 0.801·2-s − 1.35·4-s + 2.74·5-s + 2.37·7-s − 2.69·8-s + 2.20·10-s + 0.250·11-s + 2.61·13-s + 1.90·14-s + 0.558·16-s + 0.293·17-s − 2.78·19-s − 3.73·20-s + 0.200·22-s + 6.68·23-s + 2.56·25-s + 2.09·26-s − 3.22·28-s + 0.355·29-s + 2.76·31-s + 5.82·32-s + 0.235·34-s + 6.53·35-s − 6.99·37-s − 2.23·38-s − 7.40·40-s + 9.71·41-s + ⋯
L(s)  = 1  + 0.566·2-s − 0.678·4-s + 1.22·5-s + 0.898·7-s − 0.951·8-s + 0.696·10-s + 0.0754·11-s + 0.724·13-s + 0.509·14-s + 0.139·16-s + 0.0711·17-s − 0.638·19-s − 0.834·20-s + 0.0427·22-s + 1.39·23-s + 0.512·25-s + 0.410·26-s − 0.609·28-s + 0.0659·29-s + 0.496·31-s + 1.03·32-s + 0.0403·34-s + 1.10·35-s − 1.14·37-s − 0.362·38-s − 1.17·40-s + 1.51·41-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

\(L(1)\) \(\approx\) \(2.268357822\)
\(L(\frac12)\) \(\approx\) \(2.268357822\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.801T + 2T^{2} \)
5 \( 1 - 2.74T + 5T^{2} \)
7 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 - 0.250T + 11T^{2} \)
13 \( 1 - 2.61T + 13T^{2} \)
17 \( 1 - 0.293T + 17T^{2} \)
19 \( 1 + 2.78T + 19T^{2} \)
23 \( 1 - 6.68T + 23T^{2} \)
29 \( 1 - 0.355T + 29T^{2} \)
31 \( 1 - 2.76T + 31T^{2} \)
37 \( 1 + 6.99T + 37T^{2} \)
41 \( 1 - 9.71T + 41T^{2} \)
43 \( 1 - 0.260T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 + 5.97T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 1.81T + 67T^{2} \)
71 \( 1 - 0.370T + 71T^{2} \)
73 \( 1 - 5.02T + 73T^{2} \)
79 \( 1 - 0.802T + 79T^{2} \)
83 \( 1 - 2.75T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−10.47017994605221985339116531382, −9.348637072378261415799462503324, −8.893347049563682696967928243676, −7.931653211445087675152416412465, −6.56973190170137728147159506157, −5.74146236624903320690813286505, −5.02188312889748839748182524500, −4.10512819238575549410036105488, −2.76093908222073722224892930864, −1.36316394305759133768250012775, 1.36316394305759133768250012775, 2.76093908222073722224892930864, 4.10512819238575549410036105488, 5.02188312889748839748182524500, 5.74146236624903320690813286505, 6.56973190170137728147159506157, 7.931653211445087675152416412465, 8.893347049563682696967928243676, 9.348637072378261415799462503324, 10.47017994605221985339116531382

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