I am not sure I understand your question. The functional equation $f(s)=\epsilon f(1-s)$ applied twice yields $\epsilon^2=1$, hence $\epsilon=\pm 1$.

Now, finding the root number $\epsilon$ of a self-dual modular $L$-function can be a tricky business, and the meaning of "finding" is of course subjective. It is known that 
$$ \epsilon =i^k\eta, $$
where $k$ is the weight of the underlying modular form (i.e. $k=2$ for a rational elliptic curve), and $\eta=\pm 1$ is the eigenvalue of the Atkin-Lehner involution on the form. If the level $N$ (i.e. the conductor for a rational elliptic curve) is square-free, then
$$ \eta=\mu(N)\lambda(N)N^{1/2}, $$
where $\lambda(N)$ is the $N$-the Hecke eigenvalue of the form. In the case of a rational elliptic curve $E$, one can readily express $\lambda(N)$ from the number of points on $E$ over $\mathbb{F}_p$ for the various primes $p$ dividing $N$. If $N$ is not square-free, then I am not aware of any way to express the root number from Hecke eigenvalues.