On $p$-adic $L$-functions for symplectic representations of GL(N) over number fields

Let $F$ be a number field, and $\pi$ a regular algebraic cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_N({\mathbb{A}}_F)$ of symplectic type. When $\pi$ is spherical at all primes $\mathfrak{p}\mathrm{\mid }p$, we construct a $p$-adic $L$-function attached to any regular non-critical spin $p$-refinement $\tilde{\pi }$ of $\pi$ to $Q$-parahoric level, where $Q$ is the $(n,n)$-parabolic. More precisely, we construct a distribution $L_p(\tilde{\pi })$ on the Galois group ${\mathrm{G}\mathrm{a}\mathrm{l}}_p$ of the maximal abelian extension of $F$ unramified outside $p\mathrm{\infty }$, and show that it interpolates all the standard critical $L$-values of $\pi$ at $p$ (including, for example, cyclotomic and anticyclotomic variation when $F$ is imaginary quadratic). We show that $L_p(\tilde{\pi })$ satisfies a natural growth condition; in particular, when $\tilde{\pi }$ is ordinary, $L_p(\tilde{\pi })$ is a (bounded) measure on ${\mathrm{G}\mathrm{a}\mathrm{l}}_p$. As a corollary, when $\pi$ is unitary, has very regular weight, and is $Q$-ordinary at all $\mathfrak{p}\mathrm{\mid }p$, we deduce non-vanishing $L(\pi \times (\chi \circ N_{F\mathrm{/}\mathbb{Q}}),1\mathrm{/}2)\mathrm{\ne }0$ of the twisted central value for all but finitely many Dirichlet characters $\chi$ of $p$-power conductor.