Properties

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

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 1.50·2-s + 0.577·3-s + 1.27·4-s − 0.876·5-s + 0.870·6-s − 0.498·7-s + 0.408·8-s + 0.333·9-s − 1.32·10-s + 0.531·11-s + 0.734·12-s + 0.287·13-s − 0.751·14-s − 0.506·15-s − 0.655·16-s + 1.59·17-s + 0.502·18-s − 0.242·19-s − 1.11·20-s − 0.287·21-s + 0.800·22-s − 0.260·23-s + 0.235·24-s − 0.231·25-s + 0.435·26-s + 0.192·27-s − 0.633·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+(1 + 6.75i)) \, \Gamma_{\R}(s+(1 - 6.75i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 3,\ (1 + 6.75741527775i, 1 - 6.75741527775i:\ ),\ 1)\)

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