 Euler product of a Hecke eigenform

For a normalized weight-$k$ cuspidal Hecke eigenform, the local recurrence gives ${\sum }_{r\ge 0}{a}_{p^r}{X}^r=(1-a_{p}X+p^{k-1}{X}^2{)}^{-1}$. Multiplicativity therefore gives $L(f,s)={\prod }_p(1-a_p{p}^{-s}+p^{k-1-2s}{)}^{-1}$ in a right half-plane. Each degree-two local factor packages all the prime-power Fourier coefficients.