We investigate the problem of best policy identification in discounted linear Markov Decision Processes in the fixed confidence setting under a generative model. We first derive an instance-specific lower bound on the expected number of samples required to identify an $\varepsilon$-optimal policy with probability $1-δ$. The lower bound characterizes the optimal sampling rule as the solution of an intricate non-convex optimization program, but can be used as the starting point to devise simple and near-optimal sampling rules and algorithms. We devise such algorithms. One of these exhibits a sample complexity upper bounded by ${\cal O}({\frac{d}{(\varepsilon+Δ)^2}} (\log(\frac{1}δ)+d))$ where $Δ$ denotes the minimum reward gap of sub-optimal actions and $d$ is the dimension of the feature space. This upper bound holds in the moderate-confidence regime (i.e., for all $δ$), and matches existing minimax and gap-dependent lower bounds. We extend our algorithm to episodic linear MDPs.