L-function for Dirichlet characters vs Hecke character

There is a question of conventions here. In lifting a Dirichlet character $\chi$ mod $N$ to a character $\chi_{\mathbf{A}}$ of $\mathbf{A}^\times / \mathbf{Q}^\times$, you can either

(1) arrange that the restriction of $\chi_{\mathbf{A}}$ to $\widehat{\mathbf{Z}}^\times \subset \mathbf{A}^\times$ coincides with $\chi$ under the natural map $\widehat{\mathbf{Z}}^\times \to (\mathbf{Z}/N\mathbf{Z})^\times$;

(2) arrange that $\chi_{\mathbf{A}}(\varpi_q)$, for $q \nmid N$ prime and $\varpi_q$ any uniformizer at $q$, agrees with $\chi(q)$.

Both of these would seem like pretty reasonable things to ask for, but you can't have both at once -- as you've noticed, one is the inverse of the other -- and the usual convention is (2). If you use convention (2) not convention (1), then the adelic and non-adelic definitions of the L-functions coincide.