# Properties

 Label 2-3e6-27.7-c1-0-13 Degree $2$ Conductor $729$ Sign $0.973 + 0.230i$ 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 + (−1.53 + 1.28i)4-s + (−3.06 − 2.57i)7-s + (6.57 + 2.39i)13-s + (0.694 − 3.93i)16-s + (0.5 − 0.866i)19-s + (4.69 − 1.71i)25-s + 8·28-s + (8.42 − 7.07i)31-s + (5 + 8.66i)37-s + (0.868 − 4.92i)43-s + (1.56 + 8.86i)49-s + (−13.1 + 4.78i)52-s + (−0.766 − 0.642i)61-s + (4.00 + 6.92i)64-s + (−4.69 − 1.71i)67-s + ⋯
 L(s)  = 1 + (−0.766 + 0.642i)4-s + (−1.15 − 0.971i)7-s + (1.82 + 0.664i)13-s + (0.173 − 0.984i)16-s + (0.114 − 0.198i)19-s + (0.939 − 0.342i)25-s + 1.51·28-s + (1.51 − 1.26i)31-s + (0.821 + 1.42i)37-s + (0.132 − 0.750i)43-s + (0.223 + 1.26i)49-s + (−1.82 + 0.664i)52-s + (−0.0980 − 0.0823i)61-s + (0.500 + 0.866i)64-s + (−0.574 − 0.208i)67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$729$$    =    $$3^{6}$$ Sign: $0.973 + 0.230i$ Analytic conductor: $$5.82109$$ Root analytic conductor: $$2.41269$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{729} (163, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 729,\ (\ :1/2),\ 0.973 + 0.230i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.13653 - 0.132842i$$ $$L(\frac12)$$ $$\approx$$ $$1.13653 - 0.132842i$$ $$L(\frac{3}{2})$$ 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 + (1.53 - 1.28i)T^{2}$$
5 $$1 + (-4.69 + 1.71i)T^{2}$$
7 $$1 + (3.06 + 2.57i)T + (1.21 + 6.89i)T^{2}$$
11 $$1 + (-10.3 - 3.76i)T^{2}$$
13 $$1 + (-6.57 - 2.39i)T + (9.95 + 8.35i)T^{2}$$
17 $$1 + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (3.99 - 22.6i)T^{2}$$
29 $$1 + (22.2 - 18.6i)T^{2}$$
31 $$1 + (-8.42 + 7.07i)T + (5.38 - 30.5i)T^{2}$$
37 $$1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (31.4 + 26.3i)T^{2}$$
43 $$1 + (-0.868 + 4.92i)T + (-40.4 - 14.7i)T^{2}$$
47 $$1 + (8.16 + 46.2i)T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 + (-55.4 + 20.1i)T^{2}$$
61 $$1 + (0.766 + 0.642i)T + (10.5 + 60.0i)T^{2}$$
67 $$1 + (4.69 + 1.71i)T + (51.3 + 43.0i)T^{2}$$
71 $$1 + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-12.2 + 4.44i)T + (60.5 - 50.7i)T^{2}$$
83 $$1 + (63.5 - 53.3i)T^{2}$$
89 $$1 + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-0.868 + 4.92i)T + (-91.1 - 33.1i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$