Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \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  − 1.41·2-s + 0.999·4-s + 5-s − 1.41·10-s + 1.41·13-s − 1.00·16-s − 19-s + 0.999·20-s + 25-s − 1.99·26-s + 1.41·32-s − 1.41·37-s + 1.41·38-s + 49-s − 1.41·50-s + 1.41·52-s + 1.41·53-s − 0.999·64-s + 1.41·65-s + 1.41·67-s + 1.99·74-s − 0.999·76-s − 1.00·80-s − 95-s − 1.41·97-s − 1.41·98-s + 0.999·100-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 855,\ (0, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.6090074489\] \[L(1,\rho) \approx 0.6526318734\]

Imaginary part of the first few zeros on the critical line

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