Second order average estimates on local data of cusp forms

We specify sufficient conditions for the square modulus of the local parameters of a family of GL_n cusp forms to be bounded on average. These conditions are global in nature and are satisfied for n ≦ 4. As an application, we show that Rankin-Selberg L-functions on , for n_i ≦ 4, satisfy the standard convexity bound. Received: 21 June 2005