Motives are linear objects that lift all different cohomology theories (e.g. mixed Hodge theory on singular cohomology, Galois action on étale cohomology) of algebraic varieties at once, and their structure is controlled by algebraic cycles and $K$-theory. In this talk we will focus on mixed Tate motives (iterated extensions of the pure Tate motives $\mathbb{Q}(-n)$ for all integers $n$), and will ask the question of the explicit description of rank $2$ mixed Tate motives. This is motivated by the understanding of the special values of Dedekind zeta functions of number fields. I will explain how this also sheds light on the search for irrationality proofs for certain mathematical constants.