Let $p_1, p_2, \dotsc, p_n$ be distinct primes, and let $\epsilon_1, \epsilon_2, \dotsc, \epsilon_n$ be an arbitrary sequence of $1$ and $-1$.

There is an integer $a$ such that $\left( \frac{a}{p_1} \right) = \epsilon_1, \left( \frac{a}{p_2} \right) = \epsilon_2, \dotsc, \left( \frac{a}{p_n} \right) = \epsilon_n$, where $\left( \frac{a}{p_i} \right)$ denotes Legendre's symbol.

What can we say about the number $a$? -- I couldn't find any results for such numbers.

In particular, I am interested in bounds for the smallest possible $|a|$.

Could you recommend any papers or books on the topic?

EDIT

It's better to write "distinct odd primes" for $p_1, p_2, \dotsc, p_n$.

I have checked for several primes $< 1000$ by using computer. The smallest $a > 0$ does not seem to go beyond the product of two largest primes.