Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 19 $
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 − 4-s + 7-s + 9-s + 12-s + 2·13-s + 16-s − 19-s − 21-s − 25-s − 27-s − 28-s + 2·31-s − 36-s − 2·39-s − 2·43-s − 48-s + 49-s − 2·52-s + 57-s + 63-s − 64-s + 75-s + 76-s + 81-s + 84-s + 2·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(399\)    =    \(3 \cdot 7 \cdot 19\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 399,\ (0, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.5941442176\] \[L(1,\rho) \approx 0.7295228738\]

Imaginary part of the first few zeros on the critical line

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