Dirichlet series
| $L(s,f)$ = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.47·5-s + 0.408·6-s − 0.823·7-s + 0.353·8-s + 0.333·9-s + 1.04·10-s − 0.897·11-s + 0.288·12-s − 0.411·13-s − 0.582·14-s + 0.849·15-s + 0.250·16-s − 0.910·17-s + 0.235·18-s + 0.829·19-s + 0.735·20-s − 0.475·21-s − 0.634·22-s − 1.44·23-s + 0.204·24-s + 1.16·25-s − 0.290·26-s + 0.192·27-s − 0.411·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 6 ^{s/2} \, \Gamma_{\R}(s+6.00i) \, \Gamma_{\R}(s-6.00i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(6\) = \(2 \cdot 3\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 6,\ (6.00033540888i, -6.00033540888i:\ ),\ 1)\) |
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
Zeros not available.
Graph of the $Z$-function along the critical line
Plot not available.