Independence of Artin \(L\)-functions (Q2418041)

It has been shown by the second author [J. Reine Angew. Math. 539, 179--184 (2001; Zbl 1013.11077)] that if $L(s,\chi_i)$ ($i=1,2,\dots,r$) are Artin $L$-functions corresponding to distinct characters $\chi_i$ of a representation of the Galois group of a finite Galois extension of the rationals, then for any $m$ their derivatives $L^{(k)}(s,\chi_i)$ ($i=1,2,\dots,r; k=0,1,\dots,m)$ are linearly independent over the complex field. Now it is shown that these derivatives are linearly independent over the field $\mathfrak M_1$ of meromorphic functions of order $<1$, and if all $\chi_i$ are irreducible, then the functions $L(s,\chi_i)$ ($i=1,2,\dots,r)$ are algebraically independent over $\mathfrak M_1$. The proof utilizes a result of \textit{J. Kaczorowski} et al. [C. R. Math. Acad. Sci., Soc. R. Can. 21, No. 1, 28--32 (1999; Zbl 0929.11029)] on linear independence of elements of the Selberg class over the complex field.