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Abstract

Let $f$ be a function defined on $\mathbb{R}$ or $\mathbb{Q}_p$ and suppose that the integral defining the Mellin transform (or zeta integral) of $f$ does not converge. We can however say that $f$ has a “weak Mellin transform” $M_f(s)$ if ${\mathop{\rm Mell}(\phi \star f,s) = \mathop{\rm Mell} (\phi,s)M_f(s)}$ for all test functions $\phi$ in $C_c^\infty(\mathbb{R}^*)$ or $C_c^\infty(\mathbb{Q}_p^*)$. We show that if $f$ is of the form $f(x) = \psi\bigl(\frac a2x^2+bx\bigl)$, where $\psi$ is an additive character on $\mathbb{R}$ or $\mathbb{Q}_p$ and $a$ is invertible, then the weak Mellin transform of $f$ exists for $\Re(s) \gt 0$, satisfies a functional equation and vanishes only for $\Re(s) = 1/2$.