Q:

Rankin-Selberg convolutions of motivic L-series

Background:
Let $M_{f_i}, i=1,2$ be two modular motives associated to cusp forms
  $f_i \in S_{w_i}(\Gamma_0(N_i))$ of weight $w_i$ and level $N_i$ respectively.
  The Rankin-Selberg convolution associates an L-series
  $L(f_1\otimes f_2,s)$ to this pair of modular forms.
  In the framework of automorphic motives a natural question is 
  whether the L-series $L(M,s)$ of a motive $M$ can be represented in
  terms of modular submotives $M_{f_i}$ as $L(M,s) \stackrel{?}{=}
  L(f_1\otimes f_2,s)$.
Question:
Is there a (practical) test that can be applied to a given
(motivic) L-series as to whether it admits a Rankin-Selberg
product representation?

A:

If $L(s,M)$ is an irreducible degree 4 motivic L-function, and $L(s,\mathrm{sym}^2(M))$ has a pole, then either $M=f_1 \otimes f_2$ for a pair of distinct classical modular forms $f_1, f_2$, or $M=\mathrm{Asai}(f)$ for $f$ cuspidal on $GL_2(K)$, with $K/\mathbb{Q}$ quadratic. You can rule out the Asai case if $L(s,\mathrm{sym}^2(M)\otimes \chi)$ is entire for any nontrivial quadratic character (in particular, entire for $\chi$ the character of $K$).  If by "practical" you mean "local", then I think the answer is no. 

