# Properties

 Label 1-280-280.243-r1-0-0 Degree $1$ Conductor $280$ Sign $-0.777 + 0.629i$ Analytic cond. $30.0901$ Root an. cond. $30.0901$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.866 + 0.5i)3-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s − i·13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s + i·27-s + 29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s + i·43-s + ⋯
 L(s)  = 1 + (0.866 + 0.5i)3-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s − i·13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s + i·27-s + 29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s + i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$280$$    =    $$2^{3} \cdot 5 \cdot 7$$ Sign: $-0.777 + 0.629i$ Analytic conductor: $$30.0901$$ Root analytic conductor: $$30.0901$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{280} (243, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 280,\ (1:\ ),\ -0.777 + 0.629i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5508495101 + 1.556425609i$$ $$L(\frac12)$$ $$\approx$$ $$0.5508495101 + 1.556425609i$$ $$L(1)$$ $$\approx$$ $$1.114803552 + 0.4822781678i$$ $$L(1)$$ $$\approx$$ $$1.114803552 + 0.4822781678i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
7 $$1$$
good3 $$1 + (0.866 + 0.5i)T$$
11 $$1 + (-0.5 + 0.866i)T$$
13 $$1 - iT$$
17 $$1 + (-0.866 - 0.5i)T$$
19 $$1 + (-0.5 - 0.866i)T$$
23 $$1 + (-0.866 + 0.5i)T$$
29 $$1 + T$$
31 $$1 + (-0.5 + 0.866i)T$$
37 $$1 + (0.866 - 0.5i)T$$
41 $$1 - T$$
43 $$1 + iT$$
47 $$1 + (-0.866 + 0.5i)T$$
53 $$1 + (0.866 + 0.5i)T$$
59 $$1 + (-0.5 + 0.866i)T$$
61 $$1 + (-0.5 - 0.866i)T$$
67 $$1 + (0.866 + 0.5i)T$$
71 $$1 - T$$
73 $$1 + (0.866 + 0.5i)T$$
79 $$1 + (-0.5 - 0.866i)T$$
83 $$1 - iT$$
89 $$1 + (-0.5 - 0.866i)T$$
97 $$1 + iT$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$