Properties

Degree 2
Conductor $ 2^{2} \cdot 11 $
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 + 11-s + 15-s − 23-s + 27-s − 31-s − 33-s − 37-s + 2·47-s + 49-s + 2·53-s − 55-s − 59-s − 67-s + 69-s − 71-s − 81-s − 89-s + 93-s − 97-s + 2·103-s + 111-s − 113-s + 115-s + 121-s + 125-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(44\)    =    \(2^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 44,\ (0, 1:\ ),\ 1)$

Euler product

\[\begin{aligned} L(s,\rho) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.3157625309\] \[L(1,\rho) \approx 0.5772184494\]

Imaginary part of the first few zeros on the critical line

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