Let \(K/{\mathbb Q}\) be a finite Galois extension, \( {\mathcal G}=\text{Gal}(K/{\mathbb Q})\), \(\chi\) a character of \({\mathcal G}\), \(L(s,\chi)\) the associated Artin \(L\)-function, \(f_{\chi}\) the Artin conductor. Suppose that \(\text{Re}(\chi(g))\geq 0\) for all \(g\in {\mathcal G}\) and that for some integer \(r\), \((s-1)^rL(s,\chi)\) is entire. The author proves that \[ f_\chi\geq (6.5735)^{a_\chi}(3.9046)^{b_\chi}(0.1134)^r, \] where \(a_\chi\), \(b_\chi\) are the exponents of the Gamma factors in the completed \(L\)-function. If \(L(s,\chi)\) is entire, then \[ f_\chi\geq (4.90)^{a_\chi}(2.91)^{b_\chi}. \] If \(\chi\) is an irreducible character of degree \(n\) such that \(L(s,\chi\bar{\chi})\) is entire, then \[ f_{\chi}^{\frac 1n}\geq 4.73(1.648)^{\frac{(a_\chi-b_\chi)^2}{n^2}}e^{-(\frac{13.34}{n})^2}. \] If \(L(s,\chi\bar{\chi})\) is entire and satisfies the Riemann hypothesis, then \[ f_{\chi}^{\frac 1n}\geq 6.59 (2.163)^{\frac{(a_\chi-b_\chi)^2}{n^2}}e^{-(\frac{13278.42}{n})^2}. \]