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De Bruijn and Newman introduced a deformation of the completed Riemann zetafunction $\zeta$, and proved there is a real constant $\Lambda$ which encodesthe movement of the nontrivial zeros of $\zeta$ under the deformation. TheRiemann hypothesis is equivalent to the assertion that $\Lambda\leq 0$. Newman,however, conjectured that $\Lambda\geq 0$, remarking, "the new conjecture is aquantitative version of the dictum that the Riemann hypothesis, if true, isonly barely so." Andrade, Chang and Miller extended the machinery developed byNewman and Polya to $L$-functions for function fields. In this setting we mustconsider a modified Newman's conjecture: $\sup_{f\in\mathcal{F}} \Lambda_f \geq0$, for $\mathcal{F}$ a family of $L$-functions.  We extend their results by proving this modified Newman's conjecture forseveral families of $L$-functions. In contrast with previous work, we are ableto exhibit specific $L$-functions for which $\Lambda_D = 0$, and thereby provea stronger statement: $\max_{L\in\mathcal{F}} \Lambda_L = 0$. Using geometrictechniques, we show a certain deformed $L$-function must have a double root,which implies $\Lambda = 0$. For a different family, we construct particularelliptic curves with $p + 1$ points over $\mathbb{F}_p$. By the Weilconjectures, this has either the maximum or minimum possible number of pointsover $\mathbb{F}_{p^{2n}}$. The fact that $#E(\mathbb{F}_{p^{2n}})$ attains thebound tells us that the associated $L$-function satisfies $\Lambda = 0$.

Newman's conjecture, zeros of the L-functions, function fields