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Let C be a smooth projective curve defined over a number field and let C' bea twist of C. In this article we relate the l-adic representations attached tothe l-adic Tate modules of the Jacobians of C and C' through an Artinrepresentation. This representation induces global relations between the localfactors of the respective Hasse-Weil L-functions. We make these relationsexplicit in a particularly illustrating situation. For every Qbar-isomorphismclass of genus 2 curves defined over Q with automorphism group isomorphic toD_8 or D_{12}, except for a finite number, we choose a representative curve C/Qsuch that, for every twist C' of C satisfying some mild condition, we are ableto determine either the local factor L_p(C'/Q,T) or the productL_p(C'/Q,T)L_p(C'/Q,-T) from the local factor L_p(C/Q,T).

The twisting representation of the $L$-function of a curve