In this article, the author gives a conjectural description of the p-adic L-functions of a motive M, defined over $${\mathbb{Q}}$$ and with coefficients in a number field K, and which has good ordinary reduction at the prime p. The main definitions for the theory of the complex L-function attached to M as well as the notion of p-adic pseudo measure are recalled. The author modifies the Euler factors of both finite and infinite primes and proposes a modified version of Deligne’s period conjecture. In this way, the formulation of the existence of the p-adic L-functions attached to M and their holomorphy properties appear in a more natural setting.
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11S40 Zeta functions and $$L$$-functions 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)