The Rankin-Selberg convolution can be used to create additional L-function from just one old L-function.
In the case of Artin L-functions (which conjecturally, cover all Langlands L-functions), we start with the form
L(s,\pi) = \prod_i^n (1-p^{-s} \alpha_i(p))
and look at the Rankin-Selberg convolution L(s,\pi_1 \times \pi) = \prod_p \prod_{i,j}^n (1-\alpha_i(p) \alpha_j(p) p^{-s})^{-1}.
Observe that this new L-functions has n^2 Satake-Langlands parameters.
This new L-function, however, is imprimitive, and actually factors as
L(s,\pi_1 \times \pi_2) = L(s,\pi,\text{sym}) \cdot L(s,\pi,\wedge),
with L(s,\pi,\text{sym}) = \prod_p \prod_{i \le j} (1-\alpha_i \alpha_j p^{-s})^{-1}
and L(s,\pi,\wedge) = \prod_p \prod_{i < j} (1-\alpha_i \alpha_j p^{-s})^{-1}.
The first one has \frac 12 n(n+1) Satake-Langlands parameters and the second one \frac12 n(n-1) of the n^2 parameters of L(s,\pi_1 \times \pi_2).