Squeeze juice from lemon half into 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 base. Using serrated knife, cut off top third of artichoke. Rub all over with the remaining lemon half and add to lemon water.
Bring a medium saucepan of water to boil, add pinch salt. Drain artichokes and add them to the boiling water. Cover and simmer over moderate heat until tender, about 15 minutes. Drain artichoke and pat dry.
Meanwhile, in shall serving dish, combine the lemon juice, olive oil, garlic, capers, dill and ginger. Add articholes and toss so thoroughly coated with marinade. Cover and refrigerate artichoke, stirring occasionally for at least 4 hours or overnight.
Allow dish to return to room temp before serving.
Can be refrigerated up to 3 days.
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