Me convence

Question

Hola,-No me convence su argumento.-I am not convinced by his story. Why do we say "me convence" instead of "me convenzo" ? Do convencer verb like gustar ?

Accepted Answer

Hola Omer
Yes, the subject of this sentence is "su argumento", this is why it is "convence", not "convenzo". 
If we place the subject in the beginning you will see it more clearly.
Su argumento no me convence.
His argument doesn't convince me. ("me" being the indirect object pronoun, like with gustar)
Bear in mind we used another less literal translation though, but I translated it literally here so you can see the similarity in the sentences.
Saludos
Inma

Answer

IMHO, me is the direct object of convence and not the indirect object.

Answer

Hola Melita 
I looked into this again and the RAE says indeed that it can be considered direct object or indirect object. My mind went straight away to indirect object pronoun because the "le" is more commonly used with convencer (leísmo is accepted here) as you can see.
Have a look here on number 3. 
Gracias y saludos
Inma

Answer

If it function like the gustar verb, shouldn’t it be indirect object? Or it does not function like gustar verbs?

Answer

I don't think it functions like gustar. If one says "a mí, no me convence su argumento ",  it sure sounds strange.  If it acted like gustar, it would mean, I don't convince the arguments.

Answer

Yes it doesn’t function like gustar. My bad.