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  + 0.713·2-s − 1.18·3-s − 0.490·4-s + 1.52·5-s − 0.847·6-s + 0.360·7-s − 1.06·8-s + 0.411·9-s + 1.09·10-s + 0.0497·11-s + 0.583·12-s + 0.0436·13-s + 0.257·14-s − 1.81·15-s − 0.268·16-s + 0.242·17-s + 0.293·18-s − 0.675·19-s − 0.750·20-s − 0.428·21-s + 0.0354·22-s + 0.908·23-s + 1.26·24-s + 1.33·25-s + 0.0311·26-s + 0.699·27-s − 0.177·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.44i)) \, \Gamma_{\R}(s+(1 - 1.44i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (1 + 1.44142854502i, 1 - 1.44142854502i:\ ),\ 1)\)

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