Properties

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

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 1.06·2-s − 0.456·3-s + 0.141·4-s − 0.290·5-s + 0.487·6-s − 0.744·7-s + 0.917·8-s − 0.791·9-s + 0.310·10-s + 0.166·11-s − 0.0644·12-s − 0.586·13-s + 0.795·14-s + 0.132·15-s − 1.12·16-s + 0.570·17-s + 0.845·18-s − 0.981·19-s − 0.0410·20-s + 0.339·21-s − 0.177·22-s + 0.662·23-s − 0.418·24-s − 0.915·25-s + 0.626·26-s + 0.817·27-s − 0.105·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+(1 + 9.53i)) \, \Gamma_{\R}(s+(1 - 9.53i)) \, L(s,f)\cr=\mathstrut & -\,\Lambda(1-s,f)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 1,\ (1 + 9.53369526135i, 1 - 9.53369526135i:\ ),\ -1)\)

Euler product

\[\begin{aligned}L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line