Your birthday star is in the constellation Draco. It has the name χ (Chi) Draconis in Johann Bayer's Uranometria star catalog. It is also called 44 Draconis in the Historia Cœlestis Britannica of John Flamsteed and Edmund Halley. It is called NS 1821+7243 A in the NStars database.
It has visual magnitude 3.68 meaning that you could see this star with the naked eye in good viewing conditions. It is marked in the center of this star chart, at celestial coordinates (J2000 equinox):
Right ascension 18:21:3.4
Declination 72:43:58.2
This star is 26.2 light years away, which means that the light we see from it today set off on its journey at about the same time that you were born. Come back in a month or two and your birthday star may change, as the light from more distant stars reaches Earth.
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