Editing Dedekind zeta function (section)

==Special values==
Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field ''K''. For example, the [[class number formula|analytic class number formula]] relates the residue at ''s''&nbsp;=&nbsp;1 to the [[class number (number theory)|class number]] ''h''(''K'') of ''K'', the [[regulator of an algebraic number field|regulator]] ''R''(''K'') of ''K'', the number ''w''(''K'') of roots of unity in ''K'', the absolute discriminant of ''K'', and the number of real and complex places of ''K''. Another example is at ''s''&nbsp;=&nbsp;0 where it has a zero whose order ''r'' is equal to the [[rank of an abelian group|rank]] of the unit group of ''O''<sub>''K''</sub> and the leading term is given by

:<math>\lim_{s\rightarrow0}s^{-r}\zeta_K(s)=-\frac{h(K)R(K)}{w(K)}.</math>

It follows from the functional equation that <math>r=r_1+r_2-1</math>.
Combining the functional equation and the fact that Γ(''s'') is infinite at all integers less than or equal to zero yields that ''ζ''<sub>''K''</sub>(''s'') vanishes at all negative even integers. It even vanishes at all negative odd integers unless ''K'' is [[totally real number field|totally real]] (i.e. ''r''<sub>2</sub>&nbsp;=&nbsp;0; e.g. '''Q''' or a [[real quadratic field]]). In the totally real case, [[Carl Ludwig Siegel]] showed that ''ζ''<sub>''K''</sub>(''s'') is a non-zero rational number at negative odd integers. [[Stephen Lichtenbaum]] conjectured specific values for these rational numbers in terms of the [[algebraic K-theory]] of ''K''.