  De Bruijn and Newman introduced a deformation of the completed Riemann zeta
function $\zeta$, and proved there is a real constant $\Lambda$ which encodes
the movement of the nontrivial zeros of $\zeta$ under the deformation. The
Riemann hypothesis is equivalent to the assertion that $\Lambda\leq 0$. Newman,
however, conjectured that $\Lambda\geq 0$, remarking, "the new conjecture is a
quantitative version of the dictum that the Riemann hypothesis, if true, is
only barely so." Andrade, Chang and Miller extended the machinery developed by
Newman and Polya to $L$-functions for function fields. In this setting we must
consider a modified Newman's conjecture: $\sup_{f\in\mathcal{F}} \Lambda_f \geq
0$, for $\mathcal{F}$ a family of $L$-functions.
  We extend their results by proving this modified Newman's conjecture for
several families of $L$-functions. In contrast with previous work, we are able
to exhibit specific $L$-functions for which $\Lambda_D = 0$, and thereby prove
a stronger statement: $\max_{L\in\mathcal{F}} \Lambda_L = 0$. Using geometric
techniques, we show a certain deformed $L$-function must have a double root,
which implies $\Lambda = 0$. For a different family, we construct particular
elliptic curves with $p + 1$ points over $\mathbb{F}_p$. By the Weil
conjectures, this has either the maximum or minimum possible number of points
over $\mathbb{F}_{p^{2n}}$. The fact that $#E(\mathbb{F}_{p^{2n}})$ attains the
bound tells us that the associated $L$-function satisfies $\Lambda = 0$.
