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  − 1.03·2-s − 0.577·3-s + 0.0803·4-s − 1.07·5-s + 0.600·6-s − 0.0667·7-s + 0.955·8-s + 0.333·9-s + 1.11·10-s − 0.301·11-s − 0.0463·12-s − 0.821·13-s + 0.0694·14-s + 0.621·15-s − 1.07·16-s − 1.83·17-s − 0.346·18-s + 1.37·19-s − 0.0864·20-s + 0.0385·21-s + 0.313·22-s − 0.720·23-s − 0.551·24-s + 0.158·25-s + 0.853·26-s − 0.192·27-s − 0.00536·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.55i)) \, \Gamma_{\R}(s+(1 - 1.55i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 33,\ (1 + 1.55606576917i, 1 - 1.55606576917i:\ ),\ -1)\)

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