Computing $L$-functions and semistable reduction of superelliptic curves

We give an explicit description of the stable reduction of superelliptic curves of the form $y^n=f(x)$ at primes $\text{p}$ whose residue characteristic is prime to the exponent $n$. We then use this description to compute the local $L$-factor of the curve and the exponent of conductor at $\text{p}$.