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We provide a new description of Deligne--Beilinson cohomology for any Shimuravariety in terms of tempered currents. This is particularly useful forcomputations of regulators of motivic classes and hence to the study ofBeilinson conjectures. As an application, we construct classes in the middledegree plus one motivic cohomology of Siegel sixfolds and we compute theirimage by Beilinson higher regulator in terms of Rankin-Selberg type automorphicintegrals. Using results of Pollack and Shah, we relate the integrals tonon-critical special values of the degree $8$ Spin $L$-functions, as predictedby Beilinson conjectures.

Tempered currents and Deligne cohomology of Shimura varieties, with an application to $\mathrm{GSp}_6$