Properties

Degree 2
Conductor 68
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 − 2·13-s + 16-s + 17-s + 18-s + 25-s + 2·26-s − 32-s − 34-s − 36-s − 49-s − 50-s − 2·52-s + 2·53-s + 64-s + 68-s + 72-s + 81-s − 2·89-s + 98-s + 100-s − 2·101-s + 2·104-s − 2·106-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 68 ^{s/2} \Gamma_{\R}(s) \Gamma_{\R}(s+1) \cdot L(s,\rho)\cr =\mathstrut & \Lambda(1-s) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(68\)    =    \(2^{2} \cdot 17\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 68,\ (0, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.3327885867\] \[L(1,\rho) \approx 0.5583995415\]

Imaginary part of the first few zeros on the critical line

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