# Properties

 Degree 2 Conductor 563 Sign $-1$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41i·2-s − 3-s − 1.00·4-s + 1.41i·5-s − 1.41i·6-s + 7-s − 2.00·10-s + 1.00·12-s + 1.41i·14-s − 1.41i·15-s − 0.999·16-s − 17-s + 19-s − 1.41i·20-s − 21-s + ⋯
 L(s)  = 1 + 1.41i·2-s − 3-s − 1.00·4-s + 1.41i·5-s − 1.41i·6-s + 7-s − 2.00·10-s + 1.00·12-s + 1.41i·14-s − 1.41i·15-s − 0.999·16-s − 17-s + 19-s − 1.41i·20-s − 21-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$563$$ $$\varepsilon$$ = $-1$ motivic weight = $$0$$ character : $\chi_{563} (562, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 563,\ (\ :0),\ -1)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.6523332469$$ $$L(\frac12)$$ $$\approx$$ $$0.6523332469$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 563$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 563$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad563 $$1 + T$$
good2 $$1 - 1.41iT - T^{2}$$
3 $$1 + T + T^{2}$$
5 $$1 - 1.41iT - T^{2}$$
7 $$1 - T + T^{2}$$
11 $$1 + T^{2}$$
13 $$1 + T^{2}$$
17 $$1 + T + T^{2}$$
19 $$1 - T + T^{2}$$
23 $$1 + T + T^{2}$$
29 $$1 - T^{2}$$
31 $$1 + 1.41iT - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 - 1.41iT - T^{2}$$
47 $$1 - T + T^{2}$$
53 $$1 - 1.41iT - T^{2}$$
59 $$1 - T + T^{2}$$
61 $$1 - T + T^{2}$$
67 $$1 - T + T^{2}$$
71 $$1 - T + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - 1.41iT - T^{2}$$
83 $$1 + 1.41iT - T^{2}$$
89 $$1 + 1.41iT - T^{2}$$
97 $$1 - 1.41iT - T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}