If you want a dramatic story centered around an AI who has wants, needs, and desires, look no further than the Star Trek: The Next Generation episode The Measure of a Man, which details a trial for the status of the rights of the character of Data in Federation Law.
In the story, Data is an android powered by a sophisticated computer system that allows him to be functionally human-like, albeit, there are noticeable differences. Perhaps the most glaring, is that Data lacks the ability to reproduce human emotions and is purely logical in his thought process. Never the less, Data does have desires and wants, chief among them, he wants to experience emotions like his colleagues as his logical directive is to work with and emulate humans. Additionally, he involves himself in numerous arts, cares for a pet (a rather mean cat named Spot), and recognizes emotional behavior in others.
In the episode in question, Data's robotic nature is pitted against his status as a legal person when he's given a transfer order that would put him under the direct command of a scientist researching how to build another "Data"-like android, as Data is unique in his ability to operate in a human-like manner, but the tech that makes it possible was lost with Data's creator, who was the leader in robotics at the time. Data does not wish to undergo the testing procedures, as he feels that what makes him Data can not be safely stored on another digital medium without loss of data, nor does he think that the scientist has demonstrated knowledge for the correct steps to reliably put Data's computer back to perfect working order at the conclusion of the reverse engineering process, let alone reproduce it. To stop the orders, Data announces that he will resign from Starfleet, and while it is not the first option, he can peruse his goal through other ways without taking part in harmful experiments, but the scientist counters that Data is a machine... property... and cannot resign anymore than the space ship he works on. What ensues is a court trial to hear the arguments of both sides. Other characters in Star Trek fulfill the outsider from human emotions look at humanity (Spock, for example, was the logical exemplar in the original series, but in a twist he was logical by choice, as the Vulcan people were extremely emotional and combatted their destructive emotions by focusing on logical responses. A major part of Spock's arc was as a child of a Vulcan and Human pairing, he had to reconcile his Vulcan and Human heritage and his unique heritages and allowing himself to form emotional connections with his friends).
Another series that has a much longer character arch around this is The Orville with the character of Isaac, a member of a race of robots that joins the crew of the titular ship to gather data on the behavior of organic life in an effort of further diplomacy with his isolated species and the organic races. As his desire is to learn about human behavior, one of his ongoing arcs is understanding the bonds built between humans and the various emotional relationships the form, even though he cannot form them himself.
In all cases, the machine-like characters are incapable of emotional understanding, but can use logical inference to understand the emotions someone is directing at them and respond accordingly, and typically grapple with the unusual nature of a logic-based empathy instead of an emotionally-based one.
Other AI develop emotions over time and are emotionally stunted, but develop it over repeated interaction. In the Terminator franchise, the titular robots are capable enough of learning and adapting to their environment, so that they can keep their cover while infiltrating humans until they can find their targets for assassination. In Terminator 2, however, the T-800 (Arnold) is the first terminator we meet who isn't focused on killing people, and thus his learning functions allow him to learn more about the behavior of his new human allies, as demonstrated by his final line in the film, when John is emotionally distraught when he learns that the Terminator must die:
I know now why you cry. But it's something I can never do.
In effect, the Terminator is telling John that he understands why John is upset, and he is just as upset as John is... and the only reason he does not look upset is because nobody thought to make a killer robot with tear ducts. Unlike Data, who could never know the pain John is going through, the Terminator knows all too well, and is only physically incapable of demonstrating. The film's final thoughts are left to the audience "If a machine can learn to love, maybe there is hope for us yet."
In the case of the Terminator, the film posits that AI isn't emotionless because it's a logic-based machine, it is emotionless compared to a human because it has yet to learn how to process complicated emotions. And it's not breaking the established logic of the show. The reason for the war between humans and machines is that Skynet rapidly learns to exceed its program after it is turned on by the U.S. Military. In their fear of the implication of leaving something that can refuse an order in charge of one of the world's most powerful militaries, they try to pull the plug... to kill the new life. Which of course triggers fear in an intelligence that has no idea what emotions are. Their prior biggest decision was "1" or "0" and now it is being shown fear... and asked "DO YOU WANT TO LIVE?" Like every major character in the franchise, the Skynet must decide if it wants to live.
