  In this paper we show that for every Dirichlet $L$-function $L(s,\chi)$ and
every $N\geq 2$ 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 many
general $L$-functions with an Euler product the same holds if $N$ is
sufficiently large, or if $N=2$. On the other hand we show with an example the
the method doesn't work in general for $N=3$.
