# Properties

 Label 1-171-171.49-r0-0-0 Degree $1$ Conductor $171$ Sign $0.612 + 0.790i$ Analytic cond. $0.794120$ Root an. cond. $0.794120$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + 23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯
 L(s)  = 1 + 2-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + 23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$171$$    =    $$3^{2} \cdot 19$$ Sign: $0.612 + 0.790i$ Analytic conductor: $$0.794120$$ Root analytic conductor: $$0.794120$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{171} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 171,\ (0:\ ),\ 0.612 + 0.790i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.659927237 + 0.8131107847i$$ $$L(\frac12)$$ $$\approx$$ $$1.659927237 + 0.8131107847i$$ $$L(1)$$ $$\approx$$ $$1.614492454 + 0.4208975525i$$ $$L(1)$$ $$\approx$$ $$1.614492454 + 0.4208975525i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
19 $$1$$
good2 $$1 + T$$
5 $$1 + (-0.5 + 0.866i)T$$
7 $$1 + (-0.5 + 0.866i)T$$
11 $$1 + (-0.5 + 0.866i)T$$
13 $$1 + T$$
17 $$1 + (-0.5 - 0.866i)T$$
23 $$1 + T$$
29 $$1 + (-0.5 - 0.866i)T$$
31 $$1 + (-0.5 - 0.866i)T$$
37 $$1 + T$$
41 $$1 + (-0.5 + 0.866i)T$$
43 $$1 + T$$
47 $$1 + (-0.5 - 0.866i)T$$
53 $$1 + (-0.5 + 0.866i)T$$
59 $$1 + (-0.5 + 0.866i)T$$
61 $$1 + (-0.5 - 0.866i)T$$
67 $$1 + T$$
71 $$1 + (-0.5 - 0.866i)T$$
73 $$1 + (-0.5 - 0.866i)T$$
79 $$1 + T$$
83 $$1 + (-0.5 + 0.866i)T$$
89 $$1 + (-0.5 + 0.866i)T$$
97 $$1 + T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$