= Solution
For a <cusp form>, the <invariant norm of a modular form> $|f(\tau)|(\operatorname{Im}\tau)^{k/2}$ is bounded on the entire half-plane. It is invariant under the level group; on a truncated <fundamental region of a modular subgroup> boundedness is compactness, and near each <modular cusp> exponential decay of the <modular cusp> expansion dominates the power of height. With this bound, Fourier inversion on a horizontal interval gives
$$
|a_n|\le e^{2\pi ny}\int_0^1|f(x+iy)|\,dx\ll e^{2\pi ny}y^{-k/2}.
$$
Choosing $y=1/n$ proves the <Fourier coefficient bound for a cusp form>, $a_n=O(n^{k/2})$. Therefore $\sum|a_n|n^{-\sigma}$ converges when $\sigma>k/2+1$, and uniformly on every compact subset of that half-plane. The locally uniform limit of its holomorphic terms is holomorphic, so
$$
\boxed{L(f,s)\text{ is holomorphic for }\operatorname{Re}s>k/2+1.}
$$
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