The coefficient of Selberg Class L-function satisfy:

$a_n <M_{\epsilon} n^{\epsilon}$ (for any $\epsilon >0$) and the $a_n$ are multiplicative.

So I would like to know if it can be shown that we also have the following partial sum bounded by a constant (maybe using also other properties of $a_n$ coefficients):

$\sum_{n=1}^{2N} a_n - a_2 a_n < M$

(As $a_n$ are multiplicative, we see that some terms disappear from the sum as $a_{2k}=a_2 a_k$ for k odd, but is there a chance to have this sum bounded by a constant?)

More generally what can you advise to read on the characterisation of coefficent of Selberg Class L-function?