  If $A(z)=\sum_{n=0}^\infty a_nz^n$ and $B(z)=\sum_{n=0}^\infty b_nz^n$ are
two formal power series, with $a_n,b_n\in \mathbb{R}$, the polynomials
$(p_n)_n$ defined by the generating function $$ A(z)B(xz)=\sum_{n=0}^\infty
p_n(x)z^n $$ are called the Brenke polynomials generated by $A$ and associated
to $B$. We say that $A\in \mathcal{R}_B$ if the Brenke polynomials $(p_n)_n$
have only real zeros. Among other results, in this paper we find necessary and
sufficient conditions on $B$ such that
$\mathcal{R}_B=\mathcal{L}\text{-}\mathcal{P}$, where
$\mathcal{L}\text{-}\mathcal{P}$ denotes the Laguerre-P\'olya class (of entire
functions). These results can be considered an extension to Brenke polynomials
of the Jensen, and P\'olya and Schur characterization
$\mathcal{R}_{e^z}=\mathcal{L}\text{-}\mathcal{P}$, for Appell polynomials.
When applying our results to a relative of the Riemann zeta function, we find
new equivalencies for the Riemann Hypothesis in terms of real-rootedness of
some sequences of Brenke polynomials.
