The statements of the explicit formula for L-functions that I am aware of require the test function to be a Schwartz function (see, e.g., equation (4.11) in Section 4 of Low lying zeros of families of L-functions by Iwaniec, Luo, and Sarnak). However, test functions of the form $$ \phi(x)=\left(\frac{\sin(\pi v x)}{\pi v x}\right)^2 $$ are often used (e.g., equation (1.42) of Iwaniec-Luo-Sarnak). However, the Fourier transform of such a function, \begin{align*} \widehat{\phi}(y)=\begin{cases} \frac{1}{v}\left(1-\frac{|y|}{v}\right) & \text{ if } |y|<v \newline 0 & \text{ if } |y|\geq v, \end{cases} \end{align*} is not smooth, and therefore $\phi$ is not rapidly decreasing. In particular, $\phi$ is not a Schwartz function.

**Questions:**

Why does the explicit formula hold for these non-Schwartz test functions?

Is it only modified versions of the explicit formula (such as (10.16) in Iwaniec-Luo-Sarnak) in which there is an error term, that allow a more flexible choice of test functions?

In the case that more general test functions are permitted, can one choose the test function to be any smooth, even, absolutely integrable function with compactly supported Fourier transform?