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  + 0.869·2-s − 0.925·3-s − 0.243·4-s − 0.755·5-s − 0.804·6-s + 1.03·7-s − 1.08·8-s − 0.143·9-s − 0.657·10-s − 1.50·11-s + 0.225·12-s + 0.986·13-s + 0.896·14-s + 0.699·15-s − 0.697·16-s + 0.242·17-s − 0.124·18-s − 0.226·19-s + 0.184·20-s − 0.954·21-s − 1.31·22-s − 0.441·23-s + 1.00·24-s − 0.429·25-s + 0.858·26-s + 1.05·27-s − 0.251·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+4.03i) \, \Gamma_{\R}(s-4.03i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (4.03161585629i, -4.03161585629i:\ ),\ -1)\)

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