Suppose E is an elliptic curve over , and p > 3 is a split multiplicative prime for E. Let q = p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields K ⊂ μq∞ , q∞√m such that p remains inert in K ∩ (μq∞ ) +. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.