Counting cyclic extensions with local conditions and applications to <i>L</i>-values in the critical strip

Abstract Let $\ell $ be an odd prime. We investigate the enumeration of cyclic extensions of degree $\ell $ over $\mathbb {Q}$ subject to specified local conditions. By ordering these extensions according to their conductors, we derive an asymptotic count with a power-saving error term. As a consequence of our results, we analyze the distribution of values of L -functions associated with these extensions in the critical strip.