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Today I want to talk about a paper by Dick Gross, where he describes how to attach a motive of Artin-Tate type to a connected reductive group over a field . The motive and its -function are particularly useful in computing some adèlic integrals that occur in the trace formula. Moreover, the motive contains some other encoded information about the group. For example, if is a finite field, then the twisted dual motive yields a formula for the order of , as shown by Steinberg.

If is a local field with characteristic 0, then the -function is finite if and only if Serre’s Euler-Poincaré measure on is non-zero. Also, there is a local functional equation relating the -function of to that of its twisted dual motive.