# Properties

 Degree 2 Conductor $3^{3}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·4-s − 7-s + 5·13-s + 4·16-s − 7·19-s − 5·25-s + 2·28-s − 4·31-s + 11·37-s + 8·43-s − 6·49-s − 10·52-s − 61-s − 8·64-s + 5·67-s − 7·73-s + 14·76-s + 17·79-s − 5·91-s − 19·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s − 4·112-s + 113-s + ⋯
 L(s)  = 1 − 4-s − 0.377·7-s + 1.38·13-s + 16-s − 1.60·19-s − 25-s + 0.377·28-s − 0.718·31-s + 1.80·37-s + 1.21·43-s − 6/7·49-s − 1.38·52-s − 0.128·61-s − 64-s + 0.610·67-s − 0.819·73-s + 1.60·76-s + 1.91·79-s − 0.524·91-s − 1.92·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 0.377·112-s + 0.0940·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$27$$    =    $$3^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{27} (1, \cdot )$ Sato-Tate : $N(\mathrm{U}(1))$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 27,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.5888795834$$ $$L(\frac12)$$ $$\approx$$ $$0.5888795834$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 3$,$F_p(T) = 1 - a_p T + p T^2 .$If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1$$
good2 $$1 + p T^{2}$$
5 $$1 + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 5 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + 7 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 11 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + T + p T^{2}$$
67 $$1 - 5 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 7 T + p T^{2}$$
79 $$1 - 17 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + 19 T + p T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}