1) The use of chi-square test is absolutely unwarranted here. Chi-square test is done to know if categorical variables are independent. The key words are 'category' and 'independence'. Category means that there is a large population, and then there are different categories to subdivide the population. Check this example from http://stattrek.com/chi-square-test/independence.aspx?Tutorial=AP

In an election survey, voters might be classified by gender (male or
female) and voting preference (Democrat, Republican, or Independent).
We could use a chi-square test for independence to determine whether
gender is related to voting preference.

As you can see, you can divide the **total** population in different categories (M/F and D/R/I). And what you interested in knowing is if those categories are independent (P<0.05) or are associated. Check the above URL again to know how the hypothesis testing (Ho and Ha) is done in case of chi-square.

Now coming to your case: Your P1, P2 and P3 data are **independent sets** and **they do not form a partition of your sample space**. What I mean is that you don't have a unique big population, out of which P1, P2 and P3 -- 3 different categories of data is drawn. In fact, the way you have posed the problem, I would assume that P1, P2 and P3 are independent. There is no point in further checking their Independence thru chi-square (even if it were right!)

2)

Now, I would like to know how is this correlation and which list are
more correlated with Z. Which analysis do you recommend to me?

I am unable to understand your need. First you wanted to check for independence and then a correlation -- these are mutually exclusive things! Also note that correlation means that you have many data of one kind (say x) and many data of other kind (say y), and you would like to know if there is a pattern in the data - means if one of them could be predicted from other. Think of it like this: if you plot all the x-y pairs, do you see a visual pattern on the graph (like increasing x increases/decreases y). Now coming to your data: you have just 2 (or 6 lets say, but then they are already independent P1, P2 and P3) numbers (=number of intersection gene with Z) - measuring different things. These 2 numbers will just form a point in the graph, that I told earlier -- there is no correlation you can draw from just only a single point in the graph.

Ok, let me guess: you are trying to give a p-value to the proportion of common genes matching with Z. I'm afraid that it is not possible. P-values are used for distributions (means a large number of objects), not for numbers. HTH! I'm running out of time :)