# Properties

 Label 1-684-684.463-r1-0-0 Degree $1$ Conductor $684$ Sign $0.846 - 0.532i$ Analytic cond. $73.5060$ Root an. cond. $73.5060$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)23-s + 25-s + 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)35-s + 37-s + 41-s + (0.5 + 0.866i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯
 L(s)  = 1 + 5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)23-s + 25-s + 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)35-s + 37-s + 41-s + (0.5 + 0.866i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$684$$    =    $$2^{2} \cdot 3^{2} \cdot 19$$ Sign: $0.846 - 0.532i$ Analytic conductor: $$73.5060$$ Root analytic conductor: $$73.5060$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{684} (463, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 684,\ (1:\ ),\ 0.846 - 0.532i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.848775231 - 0.8221380036i$$ $$L(\frac12)$$ $$\approx$$ $$2.848775231 - 0.8221380036i$$ $$L(1)$$ $$\approx$$ $$1.450728224 - 0.1758296256i$$ $$L(1)$$ $$\approx$$ $$1.450728224 - 0.1758296256i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
19 $$1$$
good5 $$1 + T$$
7 $$1 + (0.5 - 0.866i)T$$
11 $$1 + (0.5 - 0.866i)T$$
13 $$1 + (-0.5 + 0.866i)T$$
17 $$1 + (-0.5 + 0.866i)T$$
23 $$1 + (0.5 - 0.866i)T$$
29 $$1 + T$$
31 $$1 + (0.5 + 0.866i)T$$
37 $$1 + T$$
41 $$1 + 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 + (-0.5 + 0.866i)T$$
79 $$1 + (0.5 + 0.866i)T$$
83 $$1 + (0.5 - 0.866i)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}$$