# 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 $1$

# Related objects

## 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: Trivial 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}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$