Squeeze the juice from a lemon half into a large bowl of water, and add the half to the bowl. Working with 1 artichoke at a time, cut off and discard the stem. Pull off the dark outer leaves and trim any dark green patches from the base. Using a serrated knife, cut off the top third of the artichoke. Rub all over with the remaining lemon half and add to the lemon water.
Bring a medium saucepan of water to a boil and add a large pinch of salt. Drain the artichokes and add them to the boiling water. Cover and simmer over moderate heat until tender, about 15 minutes. Drain the artichokes and pat dry.
Meanwhile, in a shallow serving dish, combine the lemon juice, olive oil, garlic, capers, dill and ginger. Add the artichokes and toss so they're thoroughly coated with the marinade. Cover and refrigerate the artichokes, stirring occasionally, for at least 4 hours or overnight. Let the dish return to room temperature before serving.