Properties

Degree 4
Conductor $ 7 \cdot 19 \cdot 29 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 2-s − 5-s + 10-s − 11-s − 13-s + 17-s + 22-s − 23-s + 25-s + 26-s + 29-s − 31-s + 32-s − 34-s − 37-s + 46-s + 47-s − 50-s + 55-s − 58-s + 59-s + 62-s − 64-s + 65-s − 73-s + 74-s + 79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3857 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3857\)    =    \(7 \cdot 19 \cdot 29\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3857,\ (0, 0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.1259227638\] \[L(1,\rho) \approx 0.3653075101\]

Imaginary part of the first few zeros on the critical line

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