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In this paper we extend to the function field setting the heuristics formerlydeveloped by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integralmoments of $L$-functions. We also adapt to the function setting the heuristicsfirst developed by Conrey, Farmer and Zirnbauer to the study of mean values ofratios of $L$-functions. Specifically, the focus of this paper is on the familyof quadratic Dirichlet $L$-functions $L(s,\chi_{P})$ where the character $\chi$is defined by the Legendre symbol for polynomials in $\mathbb{F}_{q}[T]$ with$\mathbb{F}_{q}$ a finite field of odd cardinality and the averages are takenover all monic and irreducible polynomials $P$ of a given odd degree. As anapplication we also compute the formula for the one-level density for the zerosof these $L$-functions.

The Integral Moments and Ratios of Quadratic Dirichlet $L$-Functions over Monic Irreducible Polynomials in $\mathbb{F}_{q}[T]$