Properties

Degree $2$
Conductor $17$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

Learn more about

Dirichlet series

$L(s,f)$  = 1  − 1.42·2-s − 1.16·3-s + 1.02·4-s − 0.337·5-s + 1.66·6-s − 0.998·7-s − 0.0352·8-s + 0.362·9-s + 0.480·10-s − 0.312·11-s − 1.19·12-s + 1.47·13-s + 1.42·14-s + 0.393·15-s − 0.974·16-s + 0.242·17-s − 0.515·18-s − 1.54·19-s − 0.345·20-s + 1.16·21-s + 0.444·22-s + 1.61·23-s + 0.0411·24-s − 0.886·25-s − 2.09·26-s + 0.744·27-s − 1.02·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.09i)) \, \Gamma_{\R}(s+(1 - 4.09i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 17,\ (1 + 4.09967453245i, 1 - 4.09967453245i:\ ),\ 1)\)

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