Dirichlet series
| $L(s,\rho)$ = 1 | + 4-s + 5-s + 9-s + 16-s + 20-s + 25-s − 2·29-s + 36-s + 2·41-s + 45-s − 49-s − 2·59-s − 2·61-s + 64-s + 2·79-s + 80-s + 81-s + 100-s − 2·116-s + 121-s + 125-s + 144-s − 2·145-s + 2·151-s + 2·164-s − 169-s + 180-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(1205\) = \(5 \cdot 241\) |
| Sign: | $1$ |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 1205,\ (1, 1:\ ),\ 1)\) |
Particular Values
\[L(1/2,\rho) \approx 2.887952100\]
\[L(1,\rho) \approx 1.705915331\]
Euler product
\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line