I consider holomorphic modular forms $f$ of weight $k$ and level $1$. Introduce its associated $L$-function: $$L(s,f)=\sum_{n=1}^\infty \frac{\lambda_f(n)}{n^{s}}$$
It has an Euler product $\prod_p L_p(s,f). I do not have any background in L-functions, so I wonder why those local factors have the following expressions:
- $L_p(s,f) = (1-\lambda_f(p)p^{-s}+p^{-2s})^{-1}$
- $L_p(s,f) = (1-\alpha_f(p)p^{-s})^{-1}(1-\beta_f(p)p^{-s})^{-1}$
Is there always true that such two forms exist for an $L$-function? Or is it particularly attached to the modular forms?
That allows to find a relation between $\lambda_f(p)$ and $\alpha_f(p), \beta_f(p)$. What about $\lambda_f(p^2)$? Is there also a relation with the values of $\alpha_f(p^2), \beta_f(p^2)$?
It is not true that L-functions of modular forms always have this Euler product representation. This is true only for modular forms which are eigenforms for the Hecke algebra. Being an eigenform translates into identities involving the coefficients $\lambda_f(n)$ and these give rise to the Euler factors. Identities like $$\lambda_f(p^2n)-\lambda_f(p)\lambda_f(pn)+p\lambda_f(n)=0$$ are satisfied by the coefficients of eigenforms.