Properties

Degree 2
Conductor 163
Sign $unknown$
Self-dual no
Motivic weight 0

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + (0.5 − 0.866i) 2-s + (0.5 + 0.866i) 3-s + 6-s + (−0.5 + 0.866i) 7-s + 8-s + (−0.5 − 0.866i) 11-s + (0.5 + 0.866i) 14-s + (0.5 − 0.866i) 16-s + (−0.5 − 0.866i) 19-s − 21-s − 22-s + (0.5 + 0.866i) 24-s − 25-s + 27-s + (−0.5 − 0.866i) 29-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 163 ^{s/2} \Gamma_{\R}(s+1) ^{2} \cdot L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(163\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(2,\ 163,\ (1, 1:\ ),\ 0)$

Euler product

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

Particular Values

Not enough information (Dirichlet series coefficients/sign of the functional equation) to compute special values.

Imaginary part of the first few zeros on the critical line

Zeros not available.

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

Plot not available.