# Properties

 Degree $2$ Conductor $643$ Sign $-1$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 2.00·6-s + 7-s − 1.00·9-s + 1.41i·11-s − 1.41i·12-s − 1.41i·13-s + 1.41i·14-s − 0.999·16-s − 1.41i·17-s − 1.41i·18-s − 1.41i·19-s + 1.41i·21-s − 2.00·22-s + ⋯
 L(s)  = 1 + 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 2.00·6-s + 7-s − 1.00·9-s + 1.41i·11-s − 1.41i·12-s − 1.41i·13-s + 1.41i·14-s − 0.999·16-s − 1.41i·17-s − 1.41i·18-s − 1.41i·19-s + 1.41i·21-s − 2.00·22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$643$$ Sign: $-1$ Motivic weight: $$0$$ Character: $\chi_{643} (642, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 643,\ (\ :0),\ -1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9998375976$$ $$L(\frac12)$$ $$\approx$$ $$0.9998375976$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad643 $$1 + T$$
good2 $$1 - 1.41iT - T^{2}$$
3 $$1 - 1.41iT - T^{2}$$
5 $$1 - T^{2}$$
7 $$1 - T + T^{2}$$
11 $$1 - 1.41iT - T^{2}$$
13 $$1 + 1.41iT - T^{2}$$
17 $$1 + 1.41iT - T^{2}$$
19 $$1 + 1.41iT - T^{2}$$
23 $$1 + T + T^{2}$$
29 $$1 - T + T^{2}$$
31 $$1 + T + T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + 1.41iT - T^{2}$$
43 $$1 - T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + T + T^{2}$$
59 $$1 - 1.41iT - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - 1.41iT - T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - 1.41iT - T^{2}$$
79 $$1 + 1.41iT - T^{2}$$
83 $$1 + T + T^{2}$$
89 $$1 - T + T^{2}$$
97 $$1 + T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$