L-series and Mellin transforms

Write f(τ)=Σaₙexp(2πinτ) with n≥1 and set τ=iy. Each term becomes aₙexp(−2πny). Its integral against y^(s−1) is aₙΓ(s)/(2πn)^s when the real part of s is positive. To exchange the entire sum with the integral, begin farther to the right where absolute convergence is available. A standard elementary cusp-form coefficient bound is aₙ=O(n^(k/2)), so the safe half-plane Re(s)>k/2+1 suffices for this calculation.

The coefficient bound follows from boundedness of y^(k/2)|f(τ)| on the modular quotient, including its decaying cusp end, followed by Fourier extraction at height 1/n. This argument does not require the optimal Ramanujan bound. Defining L(f,s)=Σaₙn⁻ˢ, we obtain its completed version Λ(f,s)=(2π)⁻ˢΓ(s)L(f,s). There is no conductor factor at full level. A level-N completion ordinarily includes N^(s/2), and using that convention later should not change the uncompleted Dirichlet coefficients.

The modular law f(−1/τ)=τ^k f(τ) gives f(i/y)=i^k y^k f(iy). Split the Mellin integral at y=1 and substitute y=1/t in the lower piece. Because k is even, i^k is either plus or minus one and equals its own inverse. The resulting two integrals both run from one to infinity, where cusp decay controls every polynomial power of y. This representation extends the completed function holomorphically to every complex s, with locally uniform convergence on bounded sets of s.

Interchanging s and k−s in that split expression gives Λ(f,s)=i^kΛ(f,k−s). If k is congruent to two modulo four, the sign is negative and the central value must vanish. For sign plus, symmetry alone does not force a nonzero central value. Because reciprocal gamma is entire, the uncompleted L-function also extends entirely. The negative-integer zeros that compensate gamma poles arise from that continuation, rather than from substituting outside the original Dirichlet-series convergence region.

For a normalized simultaneous eigenform, coprime multiplicativity and the prime-power recurrence combine into local factors (1−aₚp⁻ˢ+p^(k−1−2s))⁻¹. Their product equals L(f,s) in a sufficiently far right half-plane, for instance the same safe range supplied by the elementary bound above. The quadratic term records the recurrence correction. For an arbitrary sum of eigenforms, the Mellin continuation and functional equation still hold at full level, but a single degree-two Euler product generally does not.

An Euler product, a functional equation, and analytic continuation are therefore related properties with different proofs and hypotheses. The Mellin argument gives the latter two directly from cusp decay and inversion; Hecke structure gives the first. At higher level the local factors at bad primes and the global root number require additional data. These distinctions become crucial when comparing a modular form with an elliptic curve: matching a few good coefficients suggests a connection, but the theorem must identify the correct level, normalization, and all local factors.