Properties

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

Related objects

Learn more about

Dirichlet series

$L(s,f)$  = 1  + 1.54·2-s + 0.246·3-s + 1.40·4-s + 0.737·5-s + 0.382·6-s − 0.261·7-s + 0.620·8-s − 0.939·9-s + 1.14·10-s − 0.953·11-s + 0.345·12-s + 0.278·13-s − 0.405·14-s + 0.181·15-s − 0.439·16-s + 1.30·17-s − 1.45·18-s + 0.0925·19-s + 1.03·20-s − 0.0645·21-s − 1.47·22-s + 1.13·23-s + 0.153·24-s − 0.456·25-s + 0.431·26-s − 0.478·27-s − 0.366·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+13.7i) \, \Gamma_{\R}(s-13.7i) \, L(s,f)\cr=\mathstrut & \,\Lambda(1-s,f)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 1,\ (13.7797513519i, -13.7797513519i:\ ),\ 1)\)

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