Prior for $\lambda$ in Poisson ($\lambda$) How is the prior in the following question $\sim \Gamma(0.5, 0.5)$?
I am relatively new to Bayesian inference and am struggling to grasp the concept of priors. 
Question:
$X \sim Poisson(\lambda)$ 
Show carefully that the conjugate prior for λ is the Gamma(α, β) distribution. For a random sample of size 10, $\sum x_i = 28$ .
Being relatively ingorant about λ you choose your prior to be $\pi(\lambda) ∼ Gamma(0.5, 0.5)$. Find the posterior distribution for λ and give its posterior mean and posterior variance. Calculate an equal-tailed 95% credible interval for λ.
I do not understand why $\pi(\lambda) ∼ Gamma(0.5, 0.5)$ is a suitable prior for $\lambda$. Any help is appreciated!

Convenience
It is quite convenient to use the gamma distribution for the prior, since the Poisson-gamma mixture is a conjugate family. This means (1) the posterior stays in the same class of distributions as the prior, only with a parameter change and (2) we can find the posterior analytically.
By Jeffrey's rule, the (improper) non-informative a priori density is given by $\pi(\lambda) = \lambda^{-1/2}$. This we may (improperly) see as the $G(\alpha,\beta)$ distribution with $\alpha = 1/2$ and $\beta \to 0$.
Why $\beta = 1/2$ then?
Choosing $\beta = 1/2$, I suppose because you are only relatively and not completely ignorant about $\lambda$ and because you want a proper prior. 
