When does the dipole caused by the motion of a two-dimensional body vanish?

The fluid velocity far from a translating body in two-dimensional irrotational flow is dipolar. This is a classical result. Here we ask when the dipolar component vanishes. Lamb (1945, \S\,\textbf{126}) provides symmetry conditions on the virtual mass tensor for this to be the case. We show that these conditions are not necessary and obtain the sufficient and necessary condition in terms of the shape of the body using conformal maps. The applicability of the condition is discussed.