Properties

Degree $2$
Conductor $17$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 1.84·2-s − 0.821·3-s + 2.38·4-s − 0.847·5-s + 1.51·6-s − 0.275·7-s − 2.54·8-s − 0.324·9-s + 1.55·10-s + 0.837·11-s − 1.96·12-s − 0.789·13-s + 0.506·14-s + 0.696·15-s + 2.30·16-s − 0.242·17-s + 0.597·18-s − 0.341·19-s − 2.02·20-s + 0.226·21-s − 1.54·22-s − 0.600·23-s + 2.09·24-s − 0.281·25-s + 1.45·26-s + 1.08·27-s − 0.656·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.96i)) \, \Gamma_{\R}(s+(1 - 1.96i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (1 + 1.96798635964i, 1 - 1.96798635964i:\ ),\ -1)\)

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