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Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\\mathfrak{p}$ of $F$ dividing $p$. Let $\\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\\mathbb{Z}} \\hspace{0.01mm} \\hspace{1mm} E(F) = 1$ and $\\#\\Big(\\underline{III}(E/F)_ {p^\\infty}\\Big) < \\infty$, then $ord_{s=1} \\ L(E/F,s) = 1$. 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