Properties

Degree $2$
Conductor $27$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

Learn more about

Dirichlet series

$L(s,f)$  = 1  − 0.901·2-s − 0.188·4-s + 0.403·5-s + 0.864·7-s + 1.07·8-s − 0.364·10-s − 0.633·11-s − 0.158·13-s − 0.779·14-s − 0.776·16-s − 0.588·17-s + 1.18·19-s − 0.0759·20-s + 0.570·22-s + 1.34·23-s − 0.836·25-s + 0.142·26-s − 0.162·28-s + 0.784·29-s − 0.630·31-s − 0.370·32-s + 0.530·34-s + 0.349·35-s − 0.134·37-s − 1.07·38-s + 0.432·40-s − 1.39·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1.45i) \, \Gamma_{\R}(s-1.45i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 27,\ (1.45034943837i, -1.45034943837i:\ ),\ 1)\)

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

Graph of the $Z$-function along the critical line