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  + 1.44·2-s + 0.0800·3-s + 1.09·4-s + 1.62·5-s + 0.115·6-s + 0.556·7-s + 0.133·8-s − 0.993·9-s + 2.34·10-s + 1.18·11-s + 0.0874·12-s − 0.838·13-s + 0.804·14-s + 0.130·15-s − 0.899·16-s − 0.242·17-s − 1.43·18-s − 0.466·19-s + 1.77·20-s + 0.0445·21-s + 1.70·22-s − 0.677·23-s + 0.0106·24-s + 1.63·25-s − 1.21·26-s − 0.159·27-s + 0.607·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+4.69i) \, \Gamma_{\R}(s-4.69i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (4.69933935i, -4.69933935i:\ ),\ 1)\)

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