Euler product of a Hecke eigenform
= Euler product of a Hecke eigenform
{c}
{title2=$L(f,s)=\prod_p(1-a_pp^{-s}+p^{k-1-2s})^{-1}$}
For a normalized weight-$k$ cuspidal <Hecke eigenform>, the local recurrence gives $\sum_{r\geq0}a_{p^r}X^r=(1-a_pX+p^{k-1}X^2)^{-1}$. Multiplicativity therefore gives $L(f,s)=\prod_p(1-a_pp^{-s}+p^{k-1-2s})^{-1}$ in a right half-plane. Each degree-two local factor packages all the prime-power <Fourier coefficients>.