Properties

Degree 4
Conductor $ 11^{2} \cdot 37 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 3-s − 5-s − 7-s − 13-s + 15-s + 16-s + 21-s + 25-s − 29-s + 35-s + 37-s + 39-s + 41-s − 47-s − 48-s + 53-s − 61-s + 65-s − 67-s − 71-s + 73-s − 75-s − 80-s + 83-s + 87-s + 91-s − 103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4477 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4477\)    =    \(11^{2} \cdot 37\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 4477,\ (0, 0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.1662902274\] \[L(1,\rho) \approx 0.4639740639\]

Imaginary part of the first few zeros on the critical line

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