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We establish generalizations of the Nyman-Beurling and B\'aez-Duarte criteriaconcerning lack of zeros of Dirichlet $L$-functions in the semi-plane $\Re(s)>1/p$ for $p\in (1,2]$. We pose and solve a natural extremal problem forDirichlet polynomials which take values one at the zeros of the corresponding$L$-function on the vertical line $\Re(s)=1/p$.

An extremal problem related to generalizations of the Nyman-Beurling and B\'aez-Duarte criteria