Zeros of linear combinations of Hecke $L$-functions on the critical line

Let us consider a linear combination of Hecke $L$-functions associated with the characters of the ideal class group of the imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$. In general, such linear combinations have many non-trivial zeros outside the critical line. Nevertheless, under certain natural conditions we shall show that the critical line is an exceptional set, which contains a large proportion of their non-trivial zeros.