We answer a challenge posed in Booker [L-functions as distributions. Math. Ann. 363(1-2) (2015), 423-454, §1.3] by proving a version of Weil's converse theorem [Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149-156] that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.