The local period integrals and essential vectors

By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of ${\rm GL}_n$ over $p$-adic fields. In each case, we show that the integrals achieve local formal $L$-functions defined by Langlands parameters, when the test vector is associated to the newform. We give the relation between local periods involving essential Whittaker functions and special values of formal $L$-factors at $s=1$ for certain distinguished or unitary representations. The period integrals are also served as standard non-zero distinguished forms.