Study of \(L(s,\chi)/\pi^s\) for \(L\)-functions relative to \(\mathbb{F}_q((T^{-1}))\) and associated to characters of degree 1 (Q1975050)

The author investigates transcendence questions related to \(L\)-functions in characteristic \(p\). Let \(k={\mathbb F}_q(T)\) be the rational function field over \({\mathbb F}_q\). \textit{L. Carlitz} [Duke Math. J. 1, 137-168 (1935; Zbl 0012.04904)] defined an analogue of \(\pi\) for the field \(k\). Let \(L(s,\chi)\) denote the Goss \(L\)-function associated to a character \(\chi\) [\textit{D. Goss}, Pac. J. Math. 105, 143-181 (1983; Zbl 0571.14010)]. In the paper under review, the author proves that there exist integer values \(s\) and characters \(\chi\) such that \(L(s,\chi)/\pi^s\) is rational, algebraic or transcendent. The case of \(L\)-functions associated to characters of level 1 is discussed in detail.