Properties

Degree $2$
Conductor $33$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 1.39·2-s − 0.577·3-s + 0.955·4-s + 0.651·5-s − 0.807·6-s − 1.28·7-s − 0.0621·8-s + 0.333·9-s + 0.911·10-s + 0.301·11-s − 0.551·12-s + 1.83·13-s − 1.79·14-s − 0.376·15-s − 1.04·16-s + 0.460·17-s + 0.466·18-s − 0.294·19-s + 0.622·20-s + 0.742·21-s + 0.421·22-s + 0.107·23-s + 0.0358·24-s − 0.575·25-s + 2.56·26-s − 0.192·27-s − 1.22·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.59i)) \, \Gamma_{\R}(s+(1 - 1.59i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 33,\ (1 + 1.59599133119i, 1 - 1.59599133119i:\ ),\ 1)\)

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