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.