# Properties

 Label 1-532-532.31-r1-0-0 Degree $1$ Conductor $532$ Sign $-0.851 - 0.524i$ Analytic cond. $57.1713$ Root an. cond. $57.1713$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s + (0.5 + 0.866i)5-s + 9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s − 17-s − 23-s + (−0.5 + 0.866i)25-s − 27-s + (0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
 L(s)  = 1 − 3-s + (0.5 + 0.866i)5-s + 9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s − 17-s − 23-s + (−0.5 + 0.866i)25-s − 27-s + (0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$532$$    =    $$2^{2} \cdot 7 \cdot 19$$ Sign: $-0.851 - 0.524i$ Analytic conductor: $$57.1713$$ Root analytic conductor: $$57.1713$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{532} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 532,\ (1:\ ),\ -0.851 - 0.524i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$-0.03864182375 + 0.1363146047i$$ $$L(\frac12)$$ $$\approx$$ $$-0.03864182375 + 0.1363146047i$$ $$L(1)$$ $$\approx$$ $$0.7035188201 + 0.1565698979i$$ $$L(1)$$ $$\approx$$ $$0.7035188201 + 0.1565698979i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1$$
19 $$1$$
good3 $$1 - T$$
5 $$1 + (0.5 + 0.866i)T$$
11 $$1 + (0.5 + 0.866i)T$$
13 $$1 + (-0.5 - 0.866i)T$$
17 $$1 - T$$
23 $$1 - T$$
29 $$1 + (0.5 + 0.866i)T$$
31 $$1 + (0.5 + 0.866i)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 + (0.5 - 0.866i)T$$
59 $$1 - T$$
61 $$1 - T$$
67 $$1 + (-0.5 + 0.866i)T$$
71 $$1 + (-0.5 + 0.866i)T$$
73 $$1 - T$$
79 $$1 + (-0.5 - 0.866i)T$$
83 $$1 + T$$
89 $$1 + T$$
97 $$1 + (-0.5 + 0.866i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$