# Properties

 Label 1-240-240.227-r1-0-0 Degree $1$ Conductor $240$ Sign $0.584 + 0.811i$ Analytic cond. $25.7915$ Root an. cond. $25.7915$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·7-s − i·11-s + 13-s − i·17-s + i·19-s + i·23-s + i·29-s − 31-s + 37-s + 41-s + 43-s + i·47-s − 49-s + 53-s + i·59-s + ⋯
 L(s)  = 1 + i·7-s − i·11-s + 13-s − i·17-s + i·19-s + i·23-s + i·29-s − 31-s + 37-s + 41-s + 43-s + i·47-s − 49-s + 53-s + i·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$240$$    =    $$2^{4} \cdot 3 \cdot 5$$ Sign: $0.584 + 0.811i$ Analytic conductor: $$25.7915$$ Root analytic conductor: $$25.7915$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{240} (227, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 240,\ (1:\ ),\ 0.584 + 0.811i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.635299016 + 0.8371394824i$$ $$L(\frac12)$$ $$\approx$$ $$1.635299016 + 0.8371394824i$$ $$L(1)$$ $$\approx$$ $$1.132334851 + 0.1837526503i$$ $$L(1)$$ $$\approx$$ $$1.132334851 + 0.1837526503i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
good7 $$1$$
11 $$1$$
13 $$1$$
17 $$1$$
19 $$1 + iT$$
23 $$1$$
29 $$1$$
31 $$1$$
37 $$1 - iT$$
41 $$1$$
43 $$1 + T$$
47 $$1$$
53 $$1$$
59 $$1$$
61 $$1 - iT$$
67 $$1$$
71 $$1 + iT$$
73 $$1$$
79 $$1$$
83 $$1$$
89 $$1 + iT$$
97 $$1$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$