Shriner then pointed the handgun at the officers, prompting at least one to open fire.
Shriner was struck by gunfire and fled into her residence, police said.
Police said the responding officers believed that she was shooting at them and returned fire.
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Jillian Shriner, 51, was hospitalized with non-life-threatening injuries, and no officers were hurt.
LAPD said a 9mm handgun was recovered from the Shriners home.
Her bail was set at $1 million; it was unclear whether she remains in custody.
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Weezer also has four other career Grammy noms.
The Shriners have been married for about 20 years and have two adopted children.
City News Service contributed to this report.
