  We present LOFAR $120-168$ MHz images of the merging galaxy cluster Abell
1240 that hosts double radio relics. In combination with the GMRT $595-629$ MHz
and VLA $2-4$ GHz data, we characterised the spectral and polarimetric
properties of the radio emission. The spectral indices for the relics steepen
from their outer edges towards the cluster centre and the electric field
vectors are approximately perpendicular to the major axes of the relics. The
results are consistent with the picture that these relics trace large-scale
shocks propagating outwards during the merger. Assuming diffusive shock
acceleration (DSA), we obtain shock Mach numbers of $\mathcal{M}=2.4$ and $2.3$
for the northern and southern shocks, respectively. For $\mathcal{M}\lesssim3$
shocks, a pre-existing population of mildly relativistic electrons is required
to explain the brightness of the relics due to the high ($>10$ per cent)
particle acceleration efficiency required. However, for $\mathcal{M}\gtrsim4$
shocks the required efficiency is $\gtrsim1\%$ and $\gtrsim0.5\%$,
respectively, which is low enough for shock acceleration directly from the
thermal pool. We used the fractional polarization to constrain the viewing
angle to $\geqslant(53\pm3)^\circ$ and $\geqslant(39\pm5)^\circ$ for the
northern and southern shocks, respectively. We found no evidence for diffuse
emission in the cluster central region. If the halo spans the entire region
between the relics ($\sim1.8\,\text{Mpc}$) our upper limit on the power is
$P_\text{1.4 GHz}=(1.4\pm0.6)\times10^{23}\,\text{W}\text{Hz}^{-1}$ which is
approximately equal to the anticipated flux from a cluster of this mass.
However, if the halo is smaller than this, our constraints on the power imply
that the halo is underluminous.
