Properties

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

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + (0.499 − 0.866i)2-s + (0.500 + 0.866i)3-s + 6-s + (−0.499 + 0.866i)7-s + 1.00·8-s + (−0.500 − 0.866i)11-s + (0.500 + 0.866i)14-s + (0.499 − 0.866i)16-s + (−0.500 − 0.866i)19-s − 21-s − 22-s + (0.500 + 0.866i)24-s − 25-s + 0.999·27-s + (−0.500 − 0.866i)29-s + ⋯

Functional equation

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

Invariants

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

Euler product

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

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.