Properties

Degree 3
Conductor $ 257^{2} $
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 + 8-s + 15-s − 19-s + 21-s + 24-s + 35-s + 37-s + 40-s − 41-s + 43-s − 47-s + 53-s + 56-s − 57-s − 61-s + 64-s − 67-s − 71-s + 81-s − 83-s − 95-s − 97-s + 101-s − 103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66049 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(66049\)    =    \(257^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 66049,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 2.886194162\] \[L(1,\rho) \approx 1.916654900\]

Imaginary part of the first few zeros on the critical line

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