I am struggling to get the right value for $Z_1$, which is $\beta^{\frac{-5}{2}}$ multiplied by constants. Once you have the value for the one particle partition function it is just about applying the definition of average energy.

$$\langle E \rangle = -\frac{ \partial \ln(Z_1)}{\partial \beta}$$

I got :

$$ \langle E \rangle = \frac{9}{4\beta}$$

The right answer is though:

$$\langle E \rangle = \frac{5}{2\beta}$$

NOTE: I know this is more a mathematical issue and I have already asked for this question in MSE, but I could not get the right value for < E > yet. If you want to have a look:

https://math.stackexchange.com/questions/2941009/solving-a-partition-function-in-polar-coordinates

More information about the partition function and more definitions if needed be. Please let me know if you need information that is not stated:

$$\beta = \frac{1}{k_B T}$$