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  + 0.0620·2-s + 0.812·3-s − 0.996·4-s − 1.79·5-s + 0.0504·6-s + 0.299·7-s − 0.123·8-s − 0.339·9-s − 0.111·10-s + 0.428·11-s − 0.809·12-s + 0.235·13-s + 0.0186·14-s − 1.45·15-s + 0.988·16-s − 0.242·17-s − 0.0210·18-s − 1.41·19-s + 1.78·20-s + 0.243·21-s + 0.0265·22-s − 0.333·23-s − 0.100·24-s + 2.22·25-s + 0.0145·26-s − 1.08·27-s − 0.298·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (1 + 4.0349843595i, 1 - 4.0349843595i:\ ),\ -1)\)

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