Pseudodifferential Operators on $${\mathbb{Q}_p}$$ and $$L$$-Series

We define a family of pseudodifferential operators on the Hilbert space $$L^2({\mathbb{Q}_p})$$ of complex valued square-integrable functions on the $$p$$ -adic number field $${\mathbb{Q}_p}$$ . These generalise the Vladimirov derivative, using the Dirichlet and other suitable multiplicative characters.The Riemann zeta-function and the related Dirichlet $$L$$ -functions can be expressed as a trace of these operators on a subspace of $$L^2({\mathbb{Q}_p})$$ . We also extend this to the $$L$$ -functions associated with modular (cusp) forms. Wavelets on $${\mathbb{Q}_p}$$ are common sets of eigenfunctions of all these operators.