Properties

Degree 2
Conductor 136
Sign $1$
Self-dual yes
Motivic weight 0

Related objects

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Normalization:  

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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

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

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

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

Particular Values

\[L(1/2,\rho) \approx 1.9464514954\] \[L(1,\rho) \approx 1.6926231907\]

Imaginary part of the first few zeros on the critical line

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