# Properties

 Label 1-65-65.49-r0-0-0 Degree $1$ Conductor $65$ Sign $0.859 - 0.511i$ 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 + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s − 12-s + 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + 18-s + (0.5 + 0.866i)19-s − 21-s + (0.5 + 0.866i)22-s + ⋯
 L(s)  = 1 + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s − 12-s + 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + 18-s + (0.5 + 0.866i)19-s − 21-s + (0.5 + 0.866i)22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$65$$    =    $$5 \cdot 13$$ Sign: $0.859 - 0.511i$ Analytic conductor: $$0.301858$$ Root analytic conductor: $$0.301858$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{65} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 65,\ (0:\ ),\ 0.859 - 0.511i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7397294789 - 0.2033022946i$$ $$L(\frac12)$$ $$\approx$$ $$0.7397294789 - 0.2033022946i$$ $$L(1)$$ $$\approx$$ $$0.8615431271 - 0.07142336024i$$ $$L(1)$$ $$\approx$$ $$0.8615431271 - 0.07142336024i$$

## Euler product

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