# Properties

 Label 1-1045-1045.999-r0-0-0 Degree $1$ Conductor $1045$ Sign $-0.889 + 0.456i$ Analytic cond. $4.85295$ Root an. cond. $4.85295$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s − 12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.913 − 0.406i)17-s + (−0.309 − 0.951i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)24-s + ⋯
 L(s)  = 1 + (0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s − 12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.913 − 0.406i)17-s + (−0.309 − 0.951i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)24-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1045$$    =    $$5 \cdot 11 \cdot 19$$ Sign: $-0.889 + 0.456i$ Analytic conductor: $$4.85295$$ Root analytic conductor: $$4.85295$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1045} (999, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1045,\ (0:\ ),\ -0.889 + 0.456i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$-0.2717321809 - 1.123953993i$$ $$L(\frac12)$$ $$\approx$$ $$-0.2717321809 - 1.123953993i$$ $$L(1)$$ $$\approx$$ $$0.7575883745 - 0.8120632944i$$ $$L(1)$$ $$\approx$$ $$0.7575883745 - 0.8120632944i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
11 $$1$$
19 $$1$$
good2 $$1 + (0.104 - 0.994i)T$$
3 $$1 + (0.978 - 0.207i)T$$
7 $$1 + (-0.309 - 0.951i)T$$
13 $$1 + (-0.913 + 0.406i)T$$
17 $$1 + (-0.913 - 0.406i)T$$
23 $$1 + (0.5 - 0.866i)T$$
29 $$1 + (-0.978 - 0.207i)T$$
31 $$1 + (-0.809 - 0.587i)T$$
37 $$1 + (-0.309 - 0.951i)T$$
41 $$1 + (-0.978 + 0.207i)T$$
43 $$1 + (0.5 + 0.866i)T$$
47 $$1 + (-0.669 - 0.743i)T$$
53 $$1 + (-0.913 + 0.406i)T$$
59 $$1 + (0.669 - 0.743i)T$$
61 $$1 + (-0.104 - 0.994i)T$$
67 $$1 + (0.5 - 0.866i)T$$
71 $$1 + (0.913 + 0.406i)T$$
73 $$1 + (-0.669 + 0.743i)T$$
79 $$1 + (-0.104 + 0.994i)T$$
83 $$1 + (0.809 - 0.587i)T$$
89 $$1 + (-0.5 + 0.866i)T$$
97 $$1 + (0.104 - 0.994i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$