# Properties

 Label 1-65-65.47-r0-0-0 Degree $1$ Conductor $65$ Sign $0.256 - 0.966i$ Analytic cond. $0.301858$ Root an. cond. $0.301858$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − i·3-s + 4-s + i·6-s + 7-s − 8-s − 9-s − i·11-s − i·12-s − 14-s + 16-s − i·17-s + 18-s − i·19-s − i·21-s + i·22-s + ⋯
 L(s)  = 1 − 2-s − i·3-s + 4-s + i·6-s + 7-s − 8-s − 9-s − i·11-s − i·12-s − 14-s + 16-s − i·17-s + 18-s − i·19-s − i·21-s + i·22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$65$$    =    $$5 \cdot 13$$ Sign: $0.256 - 0.966i$ Analytic conductor: $$0.301858$$ Root analytic conductor: $$0.301858$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{65} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 65,\ (0:\ ),\ 0.256 - 0.966i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5007106209 - 0.3850950802i$$ $$L(\frac12)$$ $$\approx$$ $$0.5007106209 - 0.3850950802i$$ $$L(1)$$ $$\approx$$ $$0.6640832105 - 0.2788922552i$$ $$L(1)$$ $$\approx$$ $$0.6640832105 - 0.2788922552i$$

## Euler product

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