Dirichlet series
| $L(s,\rho)$ = 1 | − 2-s − 3-s − 5-s + 6-s − 7-s + 10-s + 11-s + 13-s + 14-s + 15-s − 19-s + 21-s − 22-s − 26-s − 30-s + 32-s − 33-s + 35-s + 38-s − 39-s − 41-s − 42-s + 49-s − 55-s + 57-s + 59-s − 61-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s)=\mathstrut & 1609 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(1609\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(4,\ 1609,\ (0, 0, 1, 1:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,\rho) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Particular Values
\[L(1/2,\rho) \approx 0.07555864590\]
\[L(1,\rho) \approx 0.2641145321\]