Properties

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

Related objects

Learn more

Dirichlet series

$L(s,f)$  = 1  − 0.707·2-s + 1.10·3-s + 0.5·4-s + 0.904·5-s − 0.780·6-s + 0.829·7-s − 0.353·8-s + 0.218·9-s − 0.639·10-s + 0.567·11-s + 0.551·12-s − 0.828·13-s − 0.586·14-s + 0.998·15-s + 0.250·16-s − 0.310·17-s − 0.154·18-s − 0.736·19-s + 0.452·20-s + 0.915·21-s − 0.401·22-s − 0.0635·23-s − 0.390·24-s − 0.182·25-s + 0.585·26-s − 0.862·27-s + 0.414·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 2 ^{s/2} \, \Gamma_{\R}(s+8.92i) \, \Gamma_{\R}(s-8.92i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 2,\ (8.92287648699i, -8.92287648699i:\ ),\ 1)\)

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