# Properties

 Label 1-5-5.3-r1-0-0 Degree $1$ Conductor $5$ Sign $0.850 - 0.525i$ Analytic cond. $0.537324$ Root an. cond. $0.537324$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s + i·3-s − 4-s + 6-s − i·7-s + i·8-s − 9-s + 11-s − i·12-s + i·13-s − 14-s + 16-s − i·17-s + i·18-s − 19-s + ⋯
 L(s)  = 1 − i·2-s + i·3-s − 4-s + 6-s − i·7-s + i·8-s − 9-s + 11-s − i·12-s + i·13-s − 14-s + 16-s − i·17-s + i·18-s − 19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$5$$ Sign: $0.850 - 0.525i$ Analytic conductor: $$0.537324$$ Root analytic conductor: $$0.537324$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{5} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 5,\ (1:\ ),\ 0.850 - 0.525i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7637478801 - 0.2169647675i$$ $$L(\frac12)$$ $$\approx$$ $$0.7637478801 - 0.2169647675i$$ $$L(1)$$ $$\approx$$ $$0.8648062659 - 0.2041530661i$$ $$L(1)$$ $$\approx$$ $$0.8648062659 - 0.2041530661i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
good2 $$1$$
3 $$1 + T$$
7 $$1 + iT$$
11 $$1 - T$$
13 $$1$$
17 $$1 + T$$
19 $$1 - iT$$
23 $$1 + iT$$
29 $$1 - T$$
31 $$1$$
37 $$1 + T$$
41 $$1 - iT$$
43 $$1 + iT$$
47 $$1 - T$$
53 $$1$$
59 $$1 + T$$
61 $$1 - iT$$
67 $$1 + iT$$
71 $$1 - T$$
73 $$1$$
79 $$1 + T$$
83 $$1 - iT$$
89 $$1 + iT$$
97 $$1 - T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$