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  − 1.07·2-s − 0.651·3-s + 0.165·4-s − 1.08·5-s + 0.703·6-s + 0.222·7-s + 0.900·8-s − 0.575·9-s + 1.17·10-s + 0.901·11-s − 0.107·12-s − 0.674·13-s − 0.240·14-s + 0.709·15-s − 1.13·16-s − 0.242·17-s + 0.621·18-s + 1.35·19-s − 0.180·20-s − 0.145·21-s − 0.973·22-s + 0.311·23-s − 0.586·24-s + 0.186·25-s + 0.728·26-s + 1.02·27-s + 0.0369·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+4.56i) \, \Gamma_{\R}(s-4.56i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (4.5658688044i, -4.5658688044i:\ ),\ 1)\)

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