Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that $End_{\overline{L}}(E)\otimes_{\mathbf{Z}}\mathbf{Q}\simeq K$. One may associate to $E$ a Groessencharacter of $L$ taking values in $K$ $$ \psi:\mathbb{A}_L^{\times}/L^{\times}\rightarrow K^{\times}. $$,

Q1: Is there an obvious upper bound for the conductor of $\psi$ depending only on $L$ (which seems, a priori, to be almost equivalent to ask for an upper bound for the conductor of $E/L$)?

Q2: If $K$ has class number one, then we may choose $L=K$. Can we give an explicit formula for the conductor of $\psi$ in this very special situation?