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In [[number theory]], the '''local zeta function''' {{math|''Z''(''V'', ''s'')}} (sometimes called the '''congruent zeta function''' or the [[Hasse–Weil zeta function]]) is defined as
:<math>Z(V, s) = \exp\left(\sum_{
where {{mvar|V}} is a [[Singular point of an algebraic variety|non-singular]] {{mvar|n}}-dimensional [[projective algebraic variety]] over the field {{math|'''F'''<sub>''q''</sub>}} with {{mvar|q}} elements and {{math|''N''<sub>''
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:<math>
\mathit{Z} (V,u) = \exp
\left( \sum_{
</math>
as the [[formal power series]] in the variable <math>u</math>.
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</math>
:<math>
(2)\ \ \frac{d}{du} \log \mathit{Z} (V,u) = \sum_{
In other words, the local zeta function {{math|''Z''(''V'', ''u'')}} with coefficients in the [[finite field]] {{math|'''F'''<sub>''q''</sub>}} is defined as a function whose [[logarithmic derivative]] generates the number {{math|''N''<sub>''
<!--In [[number theory]], a '''local zeta function'''
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