# Properties

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

# Related objects

(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.