Properties

Degree 4
Conductor 1609
Sign $1$
Self-dual yes
Motivic weight 0

Related objects

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Normalization:  

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Dirichlet series

$L(s,\rho)$  = 1  − 2-s − 3-s − 5-s + 6-s − 7-s + 10-s + 11-s + 13-s + 14-s + 15-s − 19-s + 21-s − 22-s − 26-s − 30-s + 32-s − 33-s + 35-s + 38-s − 39-s − 41-s − 42-s + 49-s − 55-s + 57-s + 59-s − 61-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 1609 ^{s/2} \Gamma_{\R}(s) ^{2} \Gamma_{\R}(s+1) ^{2} \cdot L(s,\rho)\cr =\mathstrut & \Lambda(1-s) \end{align} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1609\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 1609,\ (0, 0, 1, 1:\ ),\ 1)$

Euler product

\[\begin{equation} L(s,\rho) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{equation}\]

Particular Values

\[L(1/2,\rho) \approx 0.0755586459\] \[L(1,\rho) \approx 0.2641145321\]

Imaginary part of the first few zeros on the critical line

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