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In [[
:<math>Z(V, s) = \exp\left(\sum_{k = 1}^\infty \frac{N_k}{k} (q^{-s})^k\right)</math>
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\left( \sum_{k=1}^{\infty} N_k \frac{t^k}{k} \right)
</math>
as the [[formal power series]] in the variable <math>
Equivalently, the local zeta function is sometimes defined as follows:
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==Motivations==
The relationship between the definitions of ''G'' and ''Z'' can be explained in a number of ways. (See for example the infinite product formula for ''Z'' below.) In practice it makes ''Z'' a [[rational function]] of ''t'', something that is interesting even in the case of ''V'' an [[elliptic curve]] over a finite field.
The local ''Z'' zeta functions are multiplied to get global ''<math>\zeta</math>'' zeta functions,
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