Throughout this question, an L-function is both an automorphic L-function and an element of the Selberg class such that whenever $F$ and $G$ are L-functions, then so are $F.G$ and $F\otimes G$, where $\otimes$ denotes the Rankin-Selberg convolution.
Let's now consider the 'kernel affine space' of an L-function $F$, defined as the affine space of minimal dimension containing all the non trivial zeroes of $F$. The analogue of RH for $F$ holds if and only if its kernel affine space is 1-dimensional.
My question is the following: as the tensor product of $R^{n}$ with $R^{m}$ is isomorphic to $R^{mn}$ can one establish that the kernel affine space of $F\otimes G$ is 1-dimensional if and only if the kernel affine spaces of both $F$ and $G$ are themselves 1-dimensional? Is a 'motivic' interpretation possible?
Thanks in advance.