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 − 0.258·5-s + 0.408·6-s + 1.66·7-s + 0.353·8-s + 0.333·9-s − 0.182·10-s + 1.06·11-s + 0.288·12-s − 0.755·13-s + 1.17·14-s − 0.149·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s + 0.0432·19-s − 0.129·20-s + 0.960·21-s + 0.753·22-s − 0.664·23-s + 0.204·24-s − 0.933·25-s − 0.534·26-s + 0.192·27-s + 0.831·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 6 ^{s/2} \, \Gamma_{\R}(s+7.90i) \, \Gamma_{\R}(s-7.90i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 6,\ (7.90114750377i, -7.90114750377i:\ ),\ 1)\)

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