Properties

Degree $2$
Conductor $1$
Sign $-1$
Primitive yes
Self-dual yes

Related objects

Learn more about

Dirichlet series

$L(s,f)$  = 1  + 0.289·2-s − 1.20·3-s − 0.916·4-s + 0.0395·5-s − 0.347·6-s + 0.448·7-s − 0.554·8-s + 0.444·9-s + 0.0114·10-s − 0.691·11-s + 1.10·12-s − 0.802·13-s + 0.129·14-s − 0.0475·15-s + 0.756·16-s − 1.03·17-s + 0.128·18-s + 0.637·19-s − 0.0362·20-s − 0.538·21-s − 0.200·22-s + 0.508·23-s + 0.666·24-s − 0.998·25-s − 0.232·26-s + 0.667·27-s − 0.410·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+(1 + 12.1i)) \, \Gamma_{\R}(s+(1 - 12.1i)) \, L(s,f)\cr=\mathstrut & -\,\Lambda(1-s,f)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 1,\ (1 + 12.1730083247i, 1 - 12.1730083247i:\ ),\ -1)\)

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