  We explicitly construct the Dirichlet series
$$L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},$$
where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$
in short Weierstrass form with Tamagawa product $m.$ Although there are no
$E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion
with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)=0.5053\dots.$ As a
corollary, we find that $L_{\mathrm{Tam}}(-1)=1.8193\dots$ is the average
Tamagawa product for elliptic curves over $\mathbb{Q}.$ We give an application
of these results to canonical and Weil heights.
