# Properties

 Degree $2$ Conductor $2008$ Sign $-0.137 + 0.990i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + (−0.5 − 1.53i)3-s + 4-s + (−0.5 − 1.53i)6-s + 8-s + (−1.30 + 0.951i)9-s + (0.190 + 0.587i)11-s + (−0.5 − 1.53i)12-s + 16-s + (0.618 − 1.90i)17-s + (−1.30 + 0.951i)18-s + (−1.61 − 1.17i)19-s + (0.190 + 0.587i)22-s + (−0.5 − 1.53i)24-s + 25-s + ⋯
 L(s)  = 1 + 2-s + (−0.5 − 1.53i)3-s + 4-s + (−0.5 − 1.53i)6-s + 8-s + (−1.30 + 0.951i)9-s + (0.190 + 0.587i)11-s + (−0.5 − 1.53i)12-s + 16-s + (0.618 − 1.90i)17-s + (−1.30 + 0.951i)18-s + (−1.61 − 1.17i)19-s + (0.190 + 0.587i)22-s + (−0.5 − 1.53i)24-s + 25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2008$$    =    $$2^{3} \cdot 251$$ Sign: $-0.137 + 0.990i$ Motivic weight: $$0$$ Character: $\chi_{2008} (1275, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2008,\ (\ :0),\ -0.137 + 0.990i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.855694768$$ $$L(\frac12)$$ $$\approx$$ $$1.855694768$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
251 $$1 + (0.809 - 0.587i)T$$
good3 $$1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}$$
5 $$1 - T^{2}$$
7 $$1 + (-0.309 - 0.951i)T^{2}$$
11 $$1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}$$
13 $$1 + (-0.309 + 0.951i)T^{2}$$
17 $$1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2}$$
19 $$1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2}$$
23 $$1 + (0.809 + 0.587i)T^{2}$$
29 $$1 + (-0.309 - 0.951i)T^{2}$$
31 $$1 + (0.809 - 0.587i)T^{2}$$
37 $$1 + (-0.309 + 0.951i)T^{2}$$
41 $$1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}$$
43 $$1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-0.309 - 0.951i)T^{2}$$
59 $$1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}$$
61 $$1 + (-0.309 - 0.951i)T^{2}$$
67 $$1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}$$
71 $$1 + (-0.309 + 0.951i)T^{2}$$
73 $$1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2}$$
79 $$1 + (-0.309 + 0.951i)T^{2}$$
83 $$1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}$$
89 $$1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}$$
97 $$1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$