Tamagawa Products of Elliptic Curves Over ℚ

We explicitly construct the Dirichlet series
$$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$
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.