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  − 0.220·2-s − 0.458·3-s − 0.951·4-s + 1.35·5-s + 0.101·6-s + 1.29·7-s + 0.431·8-s − 0.789·9-s − 0.299·10-s + 0.592·11-s + 0.436·12-s + 0.595·13-s − 0.285·14-s − 0.622·15-s + 0.855·16-s + 0.242·17-s + 0.174·18-s + 1.69·19-s − 1.29·20-s − 0.592·21-s − 0.130·22-s − 1.17·23-s − 0.197·24-s + 0.841·25-s − 0.131·26-s + 0.821·27-s − 1.22·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.55i)) \, \Gamma_{\R}(s+(1 - 4.55i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (1 + 4.55176008183i, 1 - 4.55176008183i:\ ),\ 1)\)

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