On an error term for the first moment of twisted $L$-functions

Let f be a Hecke-Maass cusp form for the full modular group and let χ be a primitive Dirichlet character modulo a prime q. Let s0 = σ0+ it0 with 1 2 ≤ σ0 < 1. We improve the error term for the first moment of L(s0, f ⊗ χ)L(s0, χ) over the family of even primitive Dirichlet characters. As an application, we show that for any t ∈ R, there exists a primitive Dirichlet character χ modulo q for which L(1/2 + it, f ⊗ χ)L(1/2 + it, χ) 6= 0 if the prime q satisfies q ≫ (1 + |t|) 543 25 .