Properties

Degree 4
Conductor $ 13 \cdot 347 $
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  − 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 − 29-s − 30-s + 32-s + 33-s + 35-s − 38-s − 39-s − 42-s − 43-s + 47-s + 55-s − 57-s + 58-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4511 ^{s/2} \, \Gamma_{\R}(s)^{3} \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4511\)    =    \(13 \cdot 347\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 4511,\ (0, 0, 0, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.05099637621\] \[L(1,\rho) \approx 0.2352141419\]

Imaginary part of the first few zeros on the critical line

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