Explore the intriguing world of exponential motives over Q in this hour-long conference talk. Delve into a new extension of Q(-1) by Q(0) that stands apart from classical motives, featuring Euler's constant in its period matrix. Examine the connection between this constant and the regularised value of Riemann's zeta function at 1. Investigate various results and open questions surrounding this extension, including its role as a "monodromy factor" for differential equations of E-functions. Learn about the collaborative research conducted with Peter Jossen, gaining insights into this fascinating area of mathematical study.