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  − 1.78·2-s + 0.577·3-s + 2.17·4-s + 1.26·5-s − 1.02·6-s + 0.629·7-s − 2.09·8-s + 0.333·9-s − 2.24·10-s − 0.301·11-s + 1.25·12-s + 1.24·13-s − 1.12·14-s + 0.727·15-s + 1.55·16-s + 0.0413·17-s − 0.593·18-s − 1.28·19-s + 2.74·20-s + 0.363·21-s + 0.537·22-s − 0.712·23-s − 1.20·24-s + 0.589·25-s − 2.21·26-s + 0.192·27-s + 1.36·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.82i)) \, \Gamma_{\R}(s+(1 - 1.82i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 33,\ (1 + 1.82341375434i, 1 - 1.82341375434i:\ ),\ 1)\)

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