Properties

Degree 3
Conductor 23
Sign $1$
Self-dual yes
Motivic weight 0

Related objects

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Normalization:  

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Dirichlet series

$\zeta_K(s)$  = 1  + 5-s + 7-s + 8-s + 11-s + 17-s + 19-s + 2·23-s + 2·25-s + 27-s + 35-s + 37-s + 40-s + 43-s + 2·49-s + 53-s + 55-s + 56-s + 3·59-s + 61-s + 64-s + 67-s + 77-s + 79-s + 83-s + 85-s + 88-s + 89-s + ⋯

Functional equation

\[\begin{align} \Lambda_K(s)=\mathstrut & 23 ^{s/2} \Gamma_{\R}(s) \Gamma_{\C}(s) \cdot \zeta_K(s)\cr =\mathstrut & \Lambda_K(1-s) \end{align} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  $(3,\ 23,\ (0:0),\ 1)$

Euler product

\[\begin{equation} \zeta_K(s) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{equation}\]

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\)\(L(s, \rho_{2.23.3t2.1c1})\)

Particular Values

\[\zeta_K(1/2) \approx -0.2541547348\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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