Properties

 Label 2-3e6-9.4-c1-0-27 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 + (−0.527 − 0.913i)2-s + (0.444 − 0.769i)4-s + (0.872 − 1.51i)5-s + (−1.22 − 2.12i)7-s − 3.04·8-s − 1.84·10-s + (0.627 + 1.08i)11-s + (2.27 − 3.93i)13-s + (−1.29 + 2.24i)14-s + (0.716 + 1.24i)16-s − 6.64·17-s + 0.249·19-s + (−0.775 − 1.34i)20-s + (0.661 − 1.14i)22-s + (−0.421 + 0.729i)23-s + ⋯
 L(s)  = 1 + (−0.372 − 0.645i)2-s + (0.222 − 0.384i)4-s + (0.390 − 0.676i)5-s + (−0.464 − 0.804i)7-s − 1.07·8-s − 0.582·10-s + (0.189 + 0.327i)11-s + (0.630 − 1.09i)13-s + (−0.346 + 0.600i)14-s + (0.179 + 0.310i)16-s − 1.61·17-s + 0.0571·19-s + (−0.173 − 0.300i)20-s + (0.140 − 0.244i)22-s + (−0.0877 + 0.152i)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: $\chi_{729} (244, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 729,\ (\ :1/2),\ -1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.03219i$$ $$L(\frac12)$$ $$\approx$$ $$1.03219i$$ $$L(\frac{3}{2})$$ 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 + (0.527 + 0.913i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + (-0.872 + 1.51i)T + (-2.5 - 4.33i)T^{2}$$
7 $$1 + (1.22 + 2.12i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + (-0.627 - 1.08i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (-2.27 + 3.93i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + 6.64T + 17T^{2}$$
19 $$1 - 0.249T + 19T^{2}$$
23 $$1 + (0.421 - 0.729i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-0.256 - 0.443i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (-0.410 + 0.710i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 - 2.60T + 37T^{2}$$
41 $$1 + (-4.07 + 7.06i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (2.16 + 3.74i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-2.65 - 4.59i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + 10.4T + 53T^{2}$$
59 $$1 + (-1.50 + 2.60i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (1.44 + 2.49i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (5.04 - 8.73i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 0.0894T + 71T^{2}$$
73 $$1 + 5.32T + 73T^{2}$$
79 $$1 + (2.38 + 4.13i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (4.02 + 6.96i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 6.70T + 89T^{2}$$
97 $$1 + (2.74 + 4.75i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$