Properties

Degree $2$
Conductor $3$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

Learn more

Dirichlet series

$L(s,f)$  = 1  − 0.0387·2-s + 0.577·3-s − 0.998·4-s − 0.948·5-s − 0.0223·6-s − 1.09·7-s + 0.0774·8-s + 0.333·9-s + 0.0367·10-s − 0.469·11-s − 0.576·12-s + 1.01·13-s + 0.0426·14-s − 0.547·15-s + 0.995·16-s − 1.53·17-s − 0.0129·18-s + 0.673·19-s + 0.947·20-s − 0.634·21-s + 0.0182·22-s − 0.578·23-s + 0.0447·24-s − 0.100·25-s − 0.0395·26-s + 0.192·27-s + 1.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+8.77i) \, \Gamma_{\R}(s-8.77i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 3,\ (8.77828239355i, -8.77828239355i:\ ),\ -1)\)

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