# Properties

 Label 2-3e6-27.16-c1-0-14 Degree $2$ Conductor $729$ Sign $-0.286 - 0.957i$ 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 + (1.87 + 1.57i)2-s + (0.694 + 3.93i)4-s + (2.30 + 0.837i)5-s + (0.347 − 1.96i)7-s + (−2.44 + 4.24i)8-s + (2.99 + 5.19i)10-s + (−2.30 + 0.837i)11-s + (−0.766 + 0.642i)13-s + (3.75 − 3.14i)14-s + (−3.75 + 1.36i)16-s + (3.67 + 6.36i)17-s + (0.5 − 0.866i)19-s + (−1.70 + 9.64i)20-s + (−5.63 − 2.05i)22-s + (−0.425 − 2.41i)23-s + ⋯
 L(s)  = 1 + (1.32 + 1.11i)2-s + (0.347 + 1.96i)4-s + (1.02 + 0.374i)5-s + (0.131 − 0.744i)7-s + (−0.866 + 1.50i)8-s + (0.948 + 1.64i)10-s + (−0.694 + 0.252i)11-s + (−0.212 + 0.178i)13-s + (1.00 − 0.841i)14-s + (−0.939 + 0.342i)16-s + (0.891 + 1.54i)17-s + (0.114 − 0.198i)19-s + (−0.380 + 2.15i)20-s + (−1.20 − 0.437i)22-s + (−0.0886 − 0.502i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$729$$    =    $$3^{6}$$ Sign: $-0.286 - 0.957i$ Analytic conductor: $$5.82109$$ Root analytic conductor: $$2.41269$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{729} (406, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 729,\ (\ :1/2),\ -0.286 - 0.957i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.10340 + 2.82535i$$ $$L(\frac12)$$ $$\approx$$ $$2.10340 + 2.82535i$$ $$L(\frac{3}{2})$$ 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 + (-1.87 - 1.57i)T + (0.347 + 1.96i)T^{2}$$
5 $$1 + (-2.30 - 0.837i)T + (3.83 + 3.21i)T^{2}$$
7 $$1 + (-0.347 + 1.96i)T + (-6.57 - 2.39i)T^{2}$$
11 $$1 + (2.30 - 0.837i)T + (8.42 - 7.07i)T^{2}$$
13 $$1 + (0.766 - 0.642i)T + (2.25 - 12.8i)T^{2}$$
17 $$1 + (-3.67 - 6.36i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (0.425 + 2.41i)T + (-21.6 + 7.86i)T^{2}$$
29 $$1 + (3.75 + 3.14i)T + (5.03 + 28.5i)T^{2}$$
31 $$1 + (0.173 + 0.984i)T + (-29.1 + 10.6i)T^{2}$$
37 $$1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-3.75 + 3.14i)T + (7.11 - 40.3i)T^{2}$$
43 $$1 + (10.3 - 3.76i)T + (32.9 - 27.6i)T^{2}$$
47 $$1 + (-1.70 + 9.64i)T + (-44.1 - 16.0i)T^{2}$$
53 $$1 - 7.34T + 53T^{2}$$
59 $$1 + (-2.30 - 0.837i)T + (45.1 + 37.9i)T^{2}$$
61 $$1 + (-0.868 + 4.92i)T + (-57.3 - 20.8i)T^{2}$$
67 $$1 + (5.36 - 4.49i)T + (11.6 - 65.9i)T^{2}$$
71 $$1 + (3.67 + 6.36i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (5.36 + 4.49i)T + (13.7 + 77.7i)T^{2}$$
83 $$1 + (9.38 + 7.87i)T + (14.4 + 81.7i)T^{2}$$
89 $$1 + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-6.57 + 2.39i)T + (74.3 - 62.3i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$