  We compute the infinite Frobenius actions on the de Rham fundamental groups
of affine curves defined over $\mathbf R$. As an application, we compute
extension classes of real mixed Hodge structures associated with the motivic
fundamental groups of affine curves. We establish connections between our
computation and special values of Rankin-Selberg L-functions, focusing on the
case of modular curves, and show that the associated extensions of mixed Hodge
structures does not split. We compute local zeta integrals not only at good
primes but also at certain bad primes.
