Dirichlet series
| $L(s,f)$ = 1 | − 1.42·3-s − 0.178·5-s − 0.631·7-s + 1.04·9-s − 1.14·11-s + 0.453·13-s + 0.255·15-s + 0.390·17-s − 1.61·19-s + 0.902·21-s + 1.02·23-s − 0.968·25-s − 0.0611·27-s − 0.288·29-s − 0.330·31-s + 1.63·33-s + 0.112·35-s − 1.41·37-s − 0.647·39-s + 0.956·41-s + 0.222·43-s − 0.186·45-s + 1.08·47-s − 0.601·49-s − 0.557·51-s − 1.23·53-s + 0.205·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+5.87i) \, \Gamma_{\R}(s-5.87i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $-1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 4,\ (5.87935415776i, -5.87935415776i:\ ),\ -1)\) |
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