Properties

Degree $2$
Conductor $6$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

Learn more

Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.64·5-s − 0.408·6-s − 1.06·7-s + 0.353·8-s + 0.333·9-s − 1.16·10-s − 0.00216·11-s − 0.288·12-s + 0.602·13-s − 0.755·14-s + 0.948·15-s + 0.250·16-s − 0.675·17-s + 0.235·18-s − 0.219·19-s − 0.821·20-s + 0.617·21-s − 0.00152·22-s − 0.361·23-s − 0.204·24-s + 1.69·25-s + 0.426·26-s − 0.192·27-s − 0.534·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 6 ^{s/2} \, \Gamma_{\R}(s+6.87i) \, \Gamma_{\R}(s-6.87i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 6,\ (6.87436376055i, -6.87436376055i:\ ),\ -1)\)

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