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  − 1.18·2-s + 0.352·3-s + 0.393·4-s + 1.53·5-s − 0.415·6-s − 0.564·7-s + 0.715·8-s − 0.875·9-s − 1.80·10-s + 0.302·11-s + 0.138·12-s + 1.29·13-s + 0.666·14-s + 0.540·15-s − 1.23·16-s − 0.242·17-s + 1.03·18-s − 0.623·19-s + 0.603·20-s − 0.199·21-s − 0.356·22-s − 0.365·23-s + 0.252·24-s + 1.34·25-s − 1.52·26-s − 0.660·27-s − 0.222·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1.84i) \, \Gamma_{\R}(s-1.84i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (1.84968790603i, -1.84968790603i:\ ),\ 1)\)

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