Dirichlet series
| $L(s,f)$ = 1 | + 1.39·2-s − 0.577·3-s + 0.955·4-s + 0.651·5-s − 0.807·6-s − 1.28·7-s − 0.0621·8-s + 0.333·9-s + 0.911·10-s + 0.301·11-s − 0.551·12-s + 1.83·13-s − 1.79·14-s − 0.376·15-s − 1.04·16-s + 0.460·17-s + 0.466·18-s − 0.294·19-s + 0.622·20-s + 0.742·21-s + 0.421·22-s + 0.107·23-s + 0.0358·24-s − 0.575·25-s + 2.56·26-s − 0.192·27-s − 1.22·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.59i)) \, \Gamma_{\R}(s+(1 - 1.59i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(33\) = \(3 \cdot 11\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 33,\ (1 + 1.59599133119i, 1 - 1.59599133119i:\ ),\ 1)\) |
Euler product
\(L(s,f) = \displaystyle\prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\)
Imaginary part of the first few zeros on the critical line