# A question about entire functions of order 1

Suppose $$f:\mathbb C \to \mathbb C$$ is an entire function on the complex plane of order $$1$$.

$$\forall\, c \in \mathbb R, \quad \lim_{t \to \pm \infty} \, f(t+ic) =0.$$
Can one conclude that $$f \equiv 0$$?
The answer is no: think of $$f(z):=\frac {\sin z} z$$. Indeed, $$\frac {\sin (t+ic)} {t+ic}=\frac {\sin(t)\cos(ic)+\cos(t)\sin(ic)} {t+ic}.$$