Replenishing p-adic L-functions and Euler systems at their bad primes

Let M denote a pure motive over a totally real number field F. If the Euler factors associated to the motive M at its bad primes satisfy a certain condition ($S^{\text{dep}}$), then we show that the (conjectural) primitive p-adic L-function attached to M is a linear combination of two S-depleted versions. We next verify ($S^{\text{dep}}$) holds for single, double and triple products of Hilbert modular forms, as well as their symmetric squares. If $F=Q$ and each ℓ-adic realisation $V_{\ell }^{\text{ét}}\left(M\right)$ is self-dual, this factorisation yields primitive zeta-elements attached to $M_{/Q^{\text{cyc}}}$ and a sharper divisibility in the Iwasawa Main Conjecture.