  We define a new class of functions, connected to the classical
Laguerre-P\'{o}lya class, which we call the shifted Laguerre-P\'{o}lya class.
Recent work of Griffin, Ono, Rolen, and Zagier shows that the Riemann Xi
function is in this class. We prove that a function being in this class is
equivalent to the Taylor coefficients, once shifted, being a degree $d$
multiplier sequence for every $d$, which is equivalent to shifted coefficients
satisfying all of the higher T\'{u}ran inequalities. This mirrors a classical
result of P\'{o}lya and Schur. We further show some order derivative of a
function in this class satisfies each extended Laguerre inequality. Finally, we
discuss some old and new conjectures about iterated inequalities for functions
in this class.
