  We study four fundamental questions about $1$-periods and give complete
answers. 1) We give a necessary and sufficient for a period integral to be
transcendental. 2) We give a qualitative description of all
$\overline{\mathbf{Q}}$-linear relations between $1$-periods, establishing
Kontsevich's period conjecture in this case. 3) Periods may vanish and we
determine all cases when this happens. 4) For a fixed $1$-motive, we derive a
general formula for the dimension of its space of periods in the spirit of
Baker's theorem.
