We show that the Novikov–Shubin invariant of an element of the integral group ring of the lamplighter group Z2≀ZZ2≀Z can be irrational. This disproves a conjecture of Lott and Lück. Furthermore we show that every positive real number is equal to the Novikov–Shubin invariant of some element of the real group ring of Z2≀ZZ2≀Z. Finally we show that the l2l2-Betti number of a matrix over the integral group ring of the group Zp≀ZZp≀Z, where pp is a natural number greater than 11, can be irrational. As such the groups Zp≀ZZp≀Z become the simplest known examples which give rise to irrational l2l2-Betti numbers.