# Properties

 Degree $2$ Conductor $287$ Sign $1$ Motivic weight $0$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.445·2-s + 1.80·3-s − 0.801·4-s − 0.801·6-s − 7-s + 0.801·8-s + 2.24·9-s − 1.44·12-s + 0.445·13-s + 0.445·14-s + 0.445·16-s − 1.24·17-s − 18-s − 1.24·19-s − 1.80·21-s − 1.80·23-s + 1.44·24-s + 25-s − 0.198·26-s + 2.24·27-s + 0.801·28-s − 32-s + 0.554·34-s − 1.80·36-s − 0.445·37-s + 0.554·38-s + 0.801·39-s + ⋯
 L(s)  = 1 − 0.445·2-s + 1.80·3-s − 0.801·4-s − 0.801·6-s − 7-s + 0.801·8-s + 2.24·9-s − 1.44·12-s + 0.445·13-s + 0.445·14-s + 0.445·16-s − 1.24·17-s − 18-s − 1.24·19-s − 1.80·21-s − 1.80·23-s + 1.44·24-s + 25-s − 0.198·26-s + 2.24·27-s + 0.801·28-s − 32-s + 0.554·34-s − 1.80·36-s − 0.445·37-s + 0.554·38-s + 0.801·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$287$$    =    $$7 \cdot 41$$ Sign: $1$ Motivic weight: $$0$$ Character: $\chi_{287} (286, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 287,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8331120063$$ $$L(\frac12)$$ $$\approx$$ $$0.8331120063$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + T$$
41 $$1 + T$$
good2 $$1 + 0.445T + T^{2}$$
3 $$1 - 1.80T + T^{2}$$
5 $$1 - T^{2}$$
11 $$1 - T^{2}$$
13 $$1 - 0.445T + T^{2}$$
17 $$1 + 1.24T + T^{2}$$
19 $$1 + 1.24T + T^{2}$$
23 $$1 + 1.80T + T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + 0.445T + T^{2}$$
43 $$1 - 1.24T + T^{2}$$
47 $$1 - 0.445T + T^{2}$$
53 $$1 - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - 0.445T + T^{2}$$
97 $$1 - 1.80T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$