If two L-functions are added together, the Euler product is destroyed.
Thus the linear combination is not an L-function, and hence we should
not expect a Riemann Hypothesis for it. This is indeed the case: Not
all the zeros of linear combinations of L-functions lie on the
critical line.
However, if the two L-functions have the same functional equation then
almost all the zeros do lie on the critical line. This is not seen
when they have different functional equations.
We will discuss these results (which are due to Bombieri and Hejhal)
during the talk, and demonstrate them using characteristic polynomials
of random unitary matrices, where similar phenomena are observed. If
the two matrices have the same determinant, almost all the zeros of
linear combinations of characteristic polynomials lie on the unit
circle, whereas if they have different determinants all the zeros lie
off the unit circle.