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  − 0.568·2-s + 1.52·3-s − 0.676·4-s + 0.748·5-s − 0.868·6-s + 1.10·7-s + 0.953·8-s + 1.33·9-s − 0.425·10-s − 0.496·11-s − 1.03·12-s + 1.33·13-s − 0.630·14-s + 1.14·15-s + 0.134·16-s − 0.242·17-s − 0.758·18-s − 1.21·19-s − 0.506·20-s + 1.69·21-s + 0.282·22-s + 0.544·23-s + 1.45·24-s − 0.440·25-s − 0.756·26-s + 0.511·27-s − 0.750·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+4.28i) \, \Gamma_{\R}(s-4.28i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (4.2850685348i, -4.2850685348i:\ ),\ 1)\)

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