Let $F$ be a local field (whose residue field is $q$) and $E$ its quadratic extension. Let $\pi$ be a irreducible principal series representation $\pi(\chi\_1, \chi\_2)$ of $GL\_F(2)$ especially where $\chi\_1, \chi\_2$ are unitary characters. Then I know $L\_F(s,\pi)=\frac{1}{1-\chi\_1(\varpi)q^s} \cdot \frac{1}{1-\chi\_2(\varpi)q^s}$. Then what is $L\_E(s,BC(\pi))$ in terms of $\chi\_1, \chi\_2$?



The effect of base change on the L-function of GL(2)?
If $\pi = \pi(\chi\_1, \chi\_2)$ is an irreducible principal series, then $BC(\pi)$ is the irreducible principal series attached to the unramified characters $\chi\_i \circ N\_{E/F}$ of $E^\times$. So if the $\chi\_i$ are unramified then we have


$$L\_E(s, BC(\pi)) = (1 - \chi\_1(\varpi)^2 q^{-2s})^{-1}(1 - \chi\_2(\varpi)^2 q^{-2s})^{-1}.$$


(The point of base change is that it corresponds to restriction of Weil-Deligne representations from $W\_F$ to $W\_E$ under the local Langlands correspondence, and this relation is completely obvious on the Galois side, since the Frobenius of $E$ is the square of the Frobenius of $F$.)


