Properties

Degree 2
Conductor $ 2^{3} \cdot 17 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 2-s + 4-s + 8-s − 9-s + 16-s − 17-s − 18-s − 25-s + 32-s − 34-s − 36-s + 2·47-s + 49-s − 50-s + 64-s − 68-s − 72-s + 81-s + 2·89-s + 2·94-s + 98-s − 100-s − 2·103-s − 121-s − 2·127-s + 128-s − 136-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(136\)    =    \(2^{3} \cdot 17\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 136,\ (1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.946451495\] \[L(1,\rho) \approx 1.692623190\]

Imaginary part of the first few zeros on the critical line

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