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In 2006, Fukaya and Kato formulated a general conjecture about $p$-adic$L$-functions for a large class of motives and derived a precise conjecturalinterpolation formula from the ETNC. We study this conjectural $p$-adic$L$-function in the setting of Hida families and compare it to the $p$-adic$L$-function constructed by Kitagawa. A priori it is not clear whether theinterpolation formulas coincide, mainly because of the rather inexplicitdefinition of periods (via comparison isomorphisms) that appear in the generalformula.  Using $p$-adic Eichler-Shimura isomorphisms and their interpolation in Hidafamilies, we compare these periods to the expressions in Kitagawa's formulacoming from modular symbols. This allows in certain cases to modify Kitagawa'sconstruction such that it produces a $p$-adic $L$-function compatible withFukaya's and Kato's conjecture.

A $p$-adic $L$-function with canonical motivic periods for Hida families