# Properties

 Label 2-171-171.94-c0-0-0 Degree $2$ Conductor $171$ Sign $-0.642 - 0.766i$ Analytic cond. $0.0853401$ Root an. cond. $0.292130$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + i·3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s − i·8-s − 9-s − 0.999i·10-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (0.866 + 0.499i)14-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)18-s + i·19-s + ⋯
 L(s)  = 1 + (−0.866 + 0.5i)2-s + i·3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s − i·8-s − 9-s − 0.999i·10-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (0.866 + 0.499i)14-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)18-s + i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$171$$    =    $$3^{2} \cdot 19$$ Sign: $-0.642 - 0.766i$ Analytic conductor: $$0.0853401$$ Root analytic conductor: $$0.292130$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{171} (94, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 171,\ (\ :0),\ -0.642 - 0.766i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3799375747$$ $$L(\frac12)$$ $$\approx$$ $$0.3799375747$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - iT$$
19 $$1 - iT$$
good2 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
5 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
7 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
11 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
13 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
17 $$1 + T^{2}$$
23 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
29 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
31 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
43 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
47 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
53 $$1 - T^{2}$$
59 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + T^{2}$$
79 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
83 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
89 $$1 - T^{2}$$
97 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$