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  + 1.86·2-s + 0.305·3-s + 2.46·4-s − 0.327·5-s + 0.568·6-s + 0.983·7-s + 2.73·8-s − 0.906·9-s − 0.609·10-s − 1.21·11-s + 0.754·12-s − 1.10·13-s + 1.83·14-s − 0.100·15-s + 2.62·16-s + 0.242·17-s − 1.68·18-s + 0.188·19-s − 0.808·20-s + 0.300·21-s − 2.26·22-s + 1.81·23-s + 0.835·24-s − 0.892·25-s − 2.05·26-s − 0.582·27-s + 2.42·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.42i)) \, \Gamma_{\R}(s+(1 - 4.42i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (1 + 4.42264984343i, 1 - 4.42264984343i:\ ),\ 1)\)

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