# Properties

 Degree $2$ Conductor $1000$ Sign $-1$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − 4-s + 0.618i·7-s − i·8-s − 9-s − 1.61·11-s + 1.61i·13-s − 0.618·14-s + 16-s − i·18-s − 0.618·19-s − 1.61i·22-s + 1.61i·23-s − 1.61·26-s − 0.618i·28-s + ⋯
 L(s)  = 1 + i·2-s − 4-s + 0.618i·7-s − i·8-s − 9-s − 1.61·11-s + 1.61i·13-s − 0.618·14-s + 16-s − i·18-s − 0.618·19-s − 1.61i·22-s + 1.61i·23-s − 1.61·26-s − 0.618i·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1000$$    =    $$2^{3} \cdot 5^{3}$$ Sign: $-1$ Motivic weight: $$0$$ Character: $\chi_{1000} (251, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1000,\ (\ :0),\ -1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5354403360$$ $$L(\frac12)$$ $$\approx$$ $$0.5354403360$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
5 $$1$$
good3 $$1 + T^{2}$$
7 $$1 - 0.618iT - T^{2}$$
11 $$1 + 1.61T + T^{2}$$
13 $$1 - 1.61iT - T^{2}$$
17 $$1 + T^{2}$$
19 $$1 + 0.618T + T^{2}$$
23 $$1 - 1.61iT - T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - 0.618iT - T^{2}$$
41 $$1 - 0.618T + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 + 1.61iT - T^{2}$$
53 $$1 + 0.618iT - T^{2}$$
59 $$1 + 0.618T + 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 - 1.61T + T^{2}$$
97 $$1 + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$