Properties

Degree 4
Conductor $ 53 \cdot 61 $
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 − 5-s − 7-s + 10-s + 14-s − 17-s + 23-s + 25-s − 31-s + 32-s + 34-s + 35-s − 46-s − 47-s + 49-s − 50-s + 53-s + 62-s − 64-s − 67-s − 70-s + 71-s − 73-s − 79-s + 81-s − 83-s + 85-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3233 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3233\)    =    \(53 \cdot 61\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3233,\ (0, 0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.1161335139\] \[L(1,\rho) \approx 0.3493362552\]

Imaginary part of the first few zeros on the critical line

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