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Let   G   be a compact   p  ‐adic Lie group and suppose that   G   is FAb, that is, that    H  /  [  H  ,  H  ]    is finite for every open subgroup   H   of   G  . The representation zeta functionζ  G    (  s  )   =   ∑   χ  ∈  Irr  (  G  )    χ    (  1  )    −  sencodes the distribution of continuous irreducible complex characters of   G  . For    p  ⩾  3  ,    it is known thatζ  G    (  s  )defines a meromorphic function on   C  .  Wedderburn's structure theorem for semisimple algebras implies thatζ  G    (  −  2  )   =   |  G  |for finite   G  . We complement this classic result by proving thatζ  G    (  −  2  )   =  0    for infinite   G  , assuming    p  ⩾  3   .

The representation zeta function of a FAb compact p ‐adic Lie group vanishes at −2