What is an L-Function and how can it be used for categorization?

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On hackernews, I came across this article, describing the LMFDB,

the database of L-functions, modular forms, and related objects

which aims at the categorization of mathematical objects where L-functions play the role of DNA.

I'm asking from a philosophical perspective, as the scope of this categorization seems enormous and intriguing ("periodic table..."). I know an L-function is a function on the complex plane, with poles, like the zeta-function. So I assume categorizations works via the distribution of these poles - but how exactly does this work here? What is it in basic terms (english, data-structure, algorithm...)?