Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ respectively. Let $v$ be a finite place of $F$. We consider the local $L$-factor (defined by Jacquet-Piateskii-Shapiro-Shalika) $L(s, \sigma_v \times \pi_v)$.

The local $L$-factors are defined starting with the local JPSS zeta integral $$ Z(s,W, W^{\prime}) = \int_{N_n \backslash \mathrm{GL}_n(F)} W \left( \begin{bmatrix} h & 0 \\ 0 & 1 \end{bmatrix} \right) W^{\prime}(h) |\det(h)|^{s-\frac{1}{2}} \mathrm{d}h $$ for Whittaker functions $W \in \mathcal{W}_{\psi}(\pi)$ and $W^{\prime} \in \mathcal{W}_{\psi}(\sigma)$.

**General question: existence of test vectors**: Can one construct Whittaker functions $W_0 \in \mathcal{W}_{\psi}(\pi), W_0^{\prime} \in \mathcal{W}_{\psi}(\sigma)$ such that
$$
Z(s,W_0, W_0^{\prime}) = L(s, \sigma_v \times \pi_v)? \quad (\star)
$$

**Answer in the unramified cases**: We know the existence of test vectors when the local representations $\sigma_v$ and $\pi_v$ are both unramified.

The test vectors are

new vectors, or another name is"essential Whittaker functions". This result may date back to Jacquet-Piateskii-Shapiro-Shalika's article "Conducteur des représentations génériques du groupe linéaire" (1981), especially the theorem on page 208.

**My question**: what about the case where $\pi_v$ or $\sigma_v$ is ramified (or both are ramified)? Is this problem still open or are there any known results?

**Some search**:

- Recently I found the article "Test vectors for Rankin–Selberg L-functions" by
Booker-Krishnamurthy-Lee, where the authors construct test vectors for the
**naive local Rankin-Selberg L-factors**through a process of unipotent averaging. However, it seems to me that the**naive local Rankin-Selberg L-factors**differ from the JPSS local $L$-factors that we want precisely at places where $\pi_v$ or $\sigma_v$ is ramified, missing the local JPSS factor associated to the ramified isobaric summands. - In the newly posted article, the author claimed in the proof of Proposition 2.10 that the existence of test vectors follows "by the definition of Rankin–Selberg L-factors" and the original article of JPSS is cited. But I cannot see how this follows directly from the definition. It seems to me that from the definition, we can only say that there exist families of finitely many Whittaker functions $\{W_i\}$ and $\{W_i^{\prime}\}$ such that $$ L(s, \sigma_v \times \pi_v) = \sum_{i} Z(s, W_i, W_i^{\prime}) $$ (e.g. page 7 of Cogdell's note)

Besides the above references, unfortunately, I found no other results of this type for ramified places.

Thank you all for your attention and sorry if the question is too naive to be on this site.