  A number of mathematical methods have been shown to model the zeroes of
$L$-functions with remarkable success, including the Ratios Conjecture and
Random Matrix Theory. In order to understand the structure of convolutions of
families of $L$-functions, we investigate how well these methods model the
zeros of such functions. Our primary focus is the convolution of the
$L$-function associated to Ramanujan's tau function with the family of
quadratic Dirichlet $L$-functions, for which J.B. Conrey and N.C. Snaith
computed the Ratios Conjecture's prediction. Our main result is performing the
number theory calculations and verifying these predictions for the one-level
density for suitably restricted test functions up to square-root error term.
Unlike Random Matrix Theory, which only predicts the main term, the Ratios
Conjecture detects the arithmetic of the family and makes detailed predictions
about their dependence in the lower order terms. Interestingly, while Random
Matrix Theory is frequently used to model behavior of L-functions (or at least
the main terms), there has been little if any work on the analogue of
convolving families of L-functions by convolving random matrix ensembles. We
explore one possibility by considering Kronecker products; unfortunately, it
appears that this is not the correct random matrix analogue to convolving
families.
