So I have never taken a class on Number Theory where L-functions would be discussed and I am learning about some things about L-functions by my own.
Say $\chi$ denote any character of the group of units modulo some integer 'b' and s $\in\mathbb{C}$ then $L(s, \chi)$ = $\sum_{n=1}^{\infty} \frac{\chi(n)}{n^{s}}$ is said to be the L-function associated to s and $\chi$.
My questions are:
Can I define a corresponding L-function as $\sum_{n=1}^{\infty} \frac{\chi(a_{n})}{(a_{n})^{s}}$ for some sequence $\{a_{n}\}_{1}^{\infty}$ $\subset\mathbb{N}$ ?
The L-functions also have the product formula as : $L(s, \chi)$ = $\sum_{n=1}^{\infty} \frac{\chi(n)}{n^{s}}$ = $\prod_{p}$ $\frac{1}{( 1-\chi(p).p^{-s})}$, $\forall$ real s $>$ 1, where product is over all primes.
Any chance of obtaining such a product formula, if (1) is possible?
Am I falling prey to naivety?
Sure, you can formally define such series, and under mild conditions about the growth of $a_n$'s, it will converge in some right-half plane. (E.g., you don't want all $a_n$'s to be 1.)
In special situations, yes. The Euler product is a key property of $L$-functions, which is why you don't want to define $L$-series with arbitrary sequences of $a_n$'s.
The two things that make the Euler product possible are: the geometric series formula and the unique factorization of integers. You should certainly understand this argument for yourself, at least formally. (Expand out each term in the product as a geometric series, and then multiply them together and collect like terms.) Then you'll see that the same argument works for special types of sequences $a_n$, e.g., if $(a_n) = (1, 3, 5, 7, ...)$, you get an Euler product but with $p$ odd.
More generally, you can think about products over subsets of primes, and what sequences they will correspond to as sums. The corresponding $L$-functions are called partial $L$-functions, because you only get part of the full Euler product. You can also replace some of the primes in the product with prime powers or suitable sets of numbers so that you have appropriate unique factorization properties. E.g., you can consider products of the form $\prod_{r \in R} \frac 1{1-\chi(r)r^{-s}}$ where $R$ is a finite or infinite set of natural numbers $> 1$ such that any two numbers in $R$ are coprime. Such products can be rewritten as sums of the form $\sum_{n \in S} \frac{\chi(n)}{n^s}$ where $S$ is the set of numbers of the form $r_1^{e_1} \dots r_k^{e_k}$ with $r_i$'s in $R$. That said, while people certainly consider partial $L$-functions, I don't know that there much is done with this more general type of $L$-series I described.