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Links:
- Unsorted/Hasse-Weil L function
- Arithmetic zeta function
- Dedekind zeta function
- L function
- Selberg zeta function
- motivic zeta function
Riemann Zeta
Completing
Adelic interpretation
Gamma factors
Analytic continuation
Counting zeros
p-adic factorization
Useful identities
Functional equation
A proof: 
Euler Product Expansion
Zeta functions appearing in the Weil Conjectures can be re-written as a ‘formal Euler product’ as follows (writing \(\left|X_{0}\right|\) for the set of closed points of \(X_{0}\), and noting that each closed point \(x_{0}\) gives \(\operatorname{deg}\left(x_{0}\right) \mathbb{F}_{q}^{d e g\left(x_{0}\right)}\)-rational points).
Rationality
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\(\zeta(1-2k)\) is rational, the proof follows from showing the Fourier coefficients in the Eisenstein series of weight \(2k\) and level 1 \(G_{2 k}(q)=\zeta(1-2 k)+2 \sum_{n \geq 1} \sigma_{2 k-1}(n) q^{n}\)
- Here \(\sigma_{2 k-1}(n)=\sum_{d \mathrel{\Big|}n} d^{2 k-1}\)
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Related to Eisenstein series:
Relation to primes
Special values
Random matrices
Relation to random matrices: 
