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For a quadratic extension $K$ of ${\mathbb Q}$, we consider orders $O$ in $K$that are not necessarily maximal and the ideal class group $Cl^+(O)$ in thenarrow sense of proper ideals of $O$. Characters of $Cl^+(O)$ of order at mosttwo are traditionally called genus characters. Explicit description of suchcharacters is known classically, but explicit $L$-functions associated to thosecharacters are only recently obtained partially by Chinta and Offen andcompletely by Kaneko and Mizuno. As remarked in the latter paper, the presentauthor also obtained the formula of such $L$-functions independently. Indeed,here we will give a simple and transparent alternative proof of the formula byrewriting explicit genus characters and their values in an adelic way startingfrom scratch. We also add an explicit formula for the genus number in the widesense, which is maybe known but rarely treated. As an appendix we give anideal-theoretic characterization of isomorphism classes of maximal orders ofthe matrix algebras $M_n(F)$ over a number field $F$ up to $GL_n(F)$ and$GL_n^+(F)$ conjugation respectively, and apply genus numbers to count themwhen $n=2$ and $F$ is quadratic. To avoid any misconception, we include someeasy known details.

Genus character $L$-functions of quadratic orders in an adelic way and maximal orders of matrix algebras