Emmy Rossum in the Peacock series 'Angelyne.'
CNN  — 

“Angelyne” approximates its subject in the wrong ways: It’s big, attention-getting and has all the depth of a billboard. Emmy Rossum stars and produced this limited series, which doesn’t profess to be truth but rather a facsimile of it. In a year that’s already given the world the superior “Pam & Tommy,” consider this evidence that impressive makeup alone isn’t enough.

Making good use of period songs to set both the time and whimsical tone (Electric Light Orchestra gets a workout), the story portrays the title character as a woman of mystery, one with big dreams who fastidiously avoids questions about her past and repeats “I’m not a woman, I’m an icon” like a mantra.

Building a shrine to Marilyn Monroe, Angelyne sings with a bar band, announces she wants to be “like Barbie” and turns one man after another into devotees, the key one for these purposes being a billboard king (Martin Freeman) who acquiesces to her wishes, even if the ultimate goal of plastering her picture all over Los Angeles remains puzzling to everyone but her.

Using the tired device of faux direct-to-camera interviews, the subject is so impenetrable at first that the focus shifts in part to an entertainment reporter (“Girls’” Alex Karpovsky) trying to pull back the curtain on who Angelyne really is.

The main problem is that the opening episodes do little to inspire the patience to stick around to find out. Rossum speaks in a honeyed voice as she disappears into the role, mastering an attention-commanding strut even before the character acquires her oversized cleavage. The only really notable wrinkle is that the characters keep questioning each other’s versions of events, although even that begins to grow stale relatively quickly.

Obviously, the timing is intended to paint Angelyne as the original “Famous for being famous” influencer/social-media star, just one who operated in an analog world. She’s also portrayed as the ultimate embodiment of a Hollywood dreamer who meticulously created her own image and transformation.

In the accompanying press notes executive producer Allison Miller addresses those issues, calling “Angelyne” “a story about Los Angeles and everything the city means to people who move here and to people who dream about moving here.”

Whether you buy into the romance of that vision or view Hollywood as a version of “The Twilight Zone,” “Angelyne” doesn’t even find a third dimension, much less the fourth.

“Angelyne” premieres May 19 on Peacock.