Properties

Degree 1
Conductor 1
Sign $1$
Self-dual yes
Motivic weight 0

Related objects

Downloads

Dirichlet series

$\zeta(s) = 1^{\mathstrut}$  + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯

Functional equation

\[\begin{align} \xi(s)=\mathstrut &\Gamma_{\R}(s) \cdot \zeta(s)\cr =\mathstrut & \xi(1-s) \end{align} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(1,\ 1,\ (0:\ ),\ 1)$

Euler product

\[\begin{equation} \zeta(s) = \prod_p (1 - p^{-s})^{-1} \end{equation}\]

Particular Values

\[\zeta(1/2) \approx -1.4603545088\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line