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 $1-s$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by David Farmer on 2016-05-15 17:17:53

**Referred to by:**

- columns.lfunc_lfunctions.analytic_normalization
- intro.tutorial
- lfunction.central_point
- lfunction.critical_line
- lfunction.critical_strip
- lfunction.label
- lfunction.motivic_weight
- lfunction.rational
- lfunction.selbergdata
- lfunction.trivial_zero
- lfunction.zeros
- rcs.rigor.lfunction.curve
- rcs.rigor.lfunction.modular
- lmfdb/lfunctions/templates/Lfunction.html (lines 30-35)

**History:**(expand/hide all)

- 2016-05-15 17:17:53 by David Farmer (Reviewed)