  We extend the work of Fouvry, Kowalski and Michel on correlation between
Hecke eigenvalues of modular forms and algebraic trace functions in order to
establish an asymptotic formula for a generalized cubic moment of modular
L-functions at the central point s = 1/2 and for prime moduli q. As an
application, we exploit our recent result on the mollification of the fourth
moment of Dirichlet L-functions to derive that for any pair
$(\omega_1,\omega_2)$ of multiplicative characters modulo q, there is a
positive proportion of $\chi$ (mod q) such that $L(\chi, 1/2 ), L(\chi\omega_1,
1/2 )$ and $L(\chi\omega_2, 1/2)$ are simultaneously not too small.
