While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form

where the sum is over the non-trivial zeros $\unicode[STIX]{x1D70C}$ of $\unicode[STIX]{x1D701}(s)$, $R(x)\in \overline{\mathbb{Q}}(x)$ is a rational function over algebraic numbers and $x>0$ is a real algebraic number. In particular, we show that the function

$$\begin{eqnarray}f(x)=\mathop{\sum }_{\unicode[STIX]{x1D70C}}\frac{x^{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}\end{eqnarray}$$

has infinitely many zeros in $(1,\infty )$, at most one of which is algebraic. The transcendence tools required for studying $f(x)$ in the range $x<1$ seem to be different from those in the range $x>1$. For $x<1$, we have the following non-vanishing theorem: If for an integer $d\geqslant 1$, $f(\unicode[STIX]{x1D70B}\sqrt{d}x)$ has a rational zero in$(0,1/\unicode[STIX]{x1D70B}\sqrt{d})$, then

$$\begin{eqnarray}L^{\prime }(1,\unicode[STIX]{x1D712}_{-d})\neq 0,\end{eqnarray}$$

where $\unicode[STIX]{x1D712}_{-d}$ is the quadratic character associated with the imaginary quadratic field $K:=\mathbb{Q}(\sqrt{-d})$. Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.