On the zeros of some Dirichlet series lying on the critical line

A linear combination of Dirichlet $L$-functions which are known not to have an Euler product is considered. It is proved that the interval
$$ \biggl[\frac12-iT,\frac12+iT\biggr] $$
contains for an arbitrary constant $c>0$ more than $cT$ zeros for $T\to\infty$.
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