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In this paper we show that for every Dirichlet $L$-function $L(s,\chi)$ andevery $N\geq 2$ the Dirichlet series $L(s,\chi)+L(2s,\chi)+\cdots+L(Ns,\chi)$have infinitely many zeros for $\sigma>1$. Moreover we show that for manygeneral $L$-functions with an Euler product the same holds if $N$ issufficiently large, or if $N=2$. On the other hand we show with an example thethe method doesn't work in general for $N=3$.

Zeros of $L(s)+L(2s)+\cdots+L(Ns)$ in the region of absolute convergence