Properties

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

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.124·5-s − 0.408·6-s + 0.306·7-s + 0.353·8-s + 0.333·9-s + 0.0880·10-s + 1.74·11-s − 0.288·12-s + 1.31·13-s + 0.216·14-s − 0.0718·15-s + 0.250·16-s − 0.516·17-s + 0.235·18-s + 0.940·19-s + 0.0622·20-s − 0.176·21-s + 1.23·22-s − 1.44·23-s − 0.204·24-s − 0.984·25-s + 0.929·26-s − 0.192·27-s + 0.153·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 6 ^{s/2} \, \Gamma_{\R}(s+(1 + 9.37i)) \, \Gamma_{\R}(s+(1 - 9.37i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 6,\ (1 + 9.37443036553i, 1 - 9.37443036553i:\ ),\ 1)\)

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