Congruences of algebraic $L$-functions of motives

We develop a framework to investigate conjectures on congruences between the algebraic part of special values of $L$-functions of congruent motives. We show that algebraic local Euler factors satisfy precise interpolation properties in $p$-adic families of motives and that algebraic $p$-adic $L$-functions exist in quite large generality for $p$-adic families of automorphic motives. We formulate two conjectures refining (and correcting) the currently existing formulation of the Equivariant Tamagawa Number Conjecture with coefficients in Hecke algebras and pointing out the links between conjectures on special values and completed cohomology.