Properties

Degree $2$
Conductor $21$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 1.59·2-s − 0.577·3-s + 1.54·4-s + 0.685·5-s + 0.920·6-s − 0.377·7-s − 0.866·8-s + 0.333·9-s − 1.09·10-s − 0.247·11-s − 0.891·12-s + 0.615·13-s + 0.602·14-s − 0.395·15-s − 0.161·16-s + 0.908·17-s − 0.531·18-s + 0.959·19-s + 1.05·20-s + 0.218·21-s + 0.395·22-s − 1.68·23-s + 0.500·24-s − 0.529·25-s − 0.981·26-s − 0.192·27-s − 0.583·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s+1.27i) \, \Gamma_{\R}(s-1.27i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 21,\ (1.27404261148i, -1.27404261148i:\ ),\ 1)\)

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