For the (d+1)-dimensional Lie group G=Z×pZp⊕d we determine through the use of p-power congruences a necessary and sufficient set of conditions whereby a collection of abelian L-functions arises from an element in K₁Zp[G]. If E is a semistable elliptic curve over Q, these abelian L-functions already exist; therefore, one can obtain many new families of higher order p-adic congruences. The first layer congruences are then verified computationally in a variety of cases.