# Properties

 Label 2-3e6-9.7-c1-0-16 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 no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.20 + 2.07i)2-s + (−1.88 − 3.26i)4-s + (0.0465 + 0.0806i)5-s + (0.289 − 0.502i)7-s + 4.24·8-s − 0.223·10-s + (−1.54 + 2.67i)11-s + (−2.10 − 3.63i)13-s + (0.696 + 1.20i)14-s + (−1.33 + 2.30i)16-s + 1.99·17-s − 3.84·19-s + (0.175 − 0.303i)20-s + (−3.71 − 6.43i)22-s + (−2.22 − 3.85i)23-s + ⋯
 L(s)  = 1 + (−0.849 + 1.47i)2-s + (−0.941 − 1.63i)4-s + (0.0208 + 0.0360i)5-s + (0.109 − 0.189i)7-s + 1.50·8-s − 0.0706·10-s + (−0.466 + 0.807i)11-s + (−0.582 − 1.00i)13-s + (0.186 + 0.322i)14-s + (−0.332 + 0.576i)16-s + 0.482·17-s − 0.882·19-s + (0.0392 − 0.0679i)20-s + (−0.791 − 1.37i)22-s + (−0.464 − 0.804i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\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: $\chi_{729} (487, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 729,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.603867$$ $$L(\frac12)$$ $$\approx$$ $$0.603867$$ $$L(\frac{3}{2})$$ 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 + (1.20 - 2.07i)T + (-1 - 1.73i)T^{2}$$
5 $$1 + (-0.0465 - 0.0806i)T + (-2.5 + 4.33i)T^{2}$$
7 $$1 + (-0.289 + 0.502i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (1.54 - 2.67i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (2.10 + 3.63i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 - 1.99T + 17T^{2}$$
19 $$1 + 3.84T + 19T^{2}$$
23 $$1 + (2.22 + 3.85i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-3.19 + 5.54i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (0.828 + 1.43i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 - 4.03T + 37T^{2}$$
41 $$1 + (0.548 + 0.949i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-3.45 + 5.97i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-1.79 + 3.11i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + 5.40T + 53T^{2}$$
59 $$1 + (5.14 + 8.90i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-6.59 + 11.4i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-4.41 - 7.65i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 1.14T + 71T^{2}$$
73 $$1 - 0.195T + 73T^{2}$$
79 $$1 + (-3.60 + 6.24i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (7.45 - 12.9i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 - 1.55T + 89T^{2}$$
97 $$1 + (2.64 - 4.58i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$