Let $f$ be a modular form (cuspidal, new, eigenform) of weight $k$ and level $N$. Let $p$ be a prime number not dividing $N$. In order to construct a $p$-adic $L$-function $L_p(f, s)$ interpolating the usual complex $L$-function, people choose a root $\alpha$ of the polynomial

$T^2-a_p X+p^{k-1}=0$

($a_p$ being the $p$-th Fourier coefficient) with the condition $v_p(\alpha)<k-1$.

Q1. Why is this necessary for the construction?

Q2. What happens when none of the roots satisfy this condtion? Is it still possible to define a p-adic L-function?