Normalization of an L-function (reviewed)

In its arithmetic normalization, an L-function $L(s)$ of weight $w$ has its central value at $s=\frac{w+1}{2}$ and the functional equation relates $s$ to $1 + w - s$. For L-functions defined by an Euler product $\prod_p L_p(s)^{-1}$ where the coefficients of $L_p$ are algebraic integers, this is the usual normalization implied by the definition.

The analytic normalization of an L-function is defined by $L_{an}(s):=L(s+w/2)$, where $L(s)$ is the L-function in its arithmetic normalization. This moves the central value to $s=1/2$, and the functional equation of $L_{an}(s)$ relates $s$ to $1-s$.