# Properties

 Degree $2$ Conductor $4000$ Sign $0.707 + 0.707i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.618i·3-s − i·7-s + 0.618·9-s − i·11-s + 0.618·13-s − 17-s − 0.618i·19-s + 0.618·21-s + i·27-s − 29-s − 1.61i·31-s + 0.618·33-s + 0.381i·39-s + 41-s − i·43-s + ⋯
 L(s)  = 1 + 0.618i·3-s − i·7-s + 0.618·9-s − i·11-s + 0.618·13-s − 17-s − 0.618i·19-s + 0.618·21-s + i·27-s − 29-s − 1.61i·31-s + 0.618·33-s + 0.381i·39-s + 41-s − i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4000$$    =    $$2^{5} \cdot 5^{3}$$ Sign: $0.707 + 0.707i$ Motivic weight: $$0$$ Character: $\chi_{4000} (2751, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4000,\ (\ :0),\ 0.707 + 0.707i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.279793340$$ $$L(\frac12)$$ $$\approx$$ $$1.279793340$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 - 0.618iT - T^{2}$$
7 $$1 + iT - T^{2}$$
11 $$1 + iT - T^{2}$$
13 $$1 - 0.618T + T^{2}$$
17 $$1 + T + T^{2}$$
19 $$1 + 0.618iT - T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + T + T^{2}$$
31 $$1 + 1.61iT - T^{2}$$
37 $$1 + T^{2}$$
41 $$1 - T + T^{2}$$
43 $$1 + iT - T^{2}$$
47 $$1 + 1.61iT - T^{2}$$
53 $$1 + 1.61T + T^{2}$$
59 $$1 - 1.61iT - T^{2}$$
61 $$1 - 0.618T + T^{2}$$
67 $$1 - 1.61iT - T^{2}$$
71 $$1 + iT - T^{2}$$
73 $$1 + 1.61T + T^{2}$$
79 $$1 + iT - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + T^{2}$$
97 $$1 - 1.61T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$