Dodgson joins the Hatters on a season-long loan, with a view to a permanent move next summer.
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"We are delighted to welcome Owen to the club. He's a player that has been on our radar for some time," County boss Dave Challinor told the club website.
"Hopefully now he can find a long-term home where he can settle, develop and really kick-start his career.
"He arrives in decent shape fitness-wise and is available to be involved immediately for us."