Properties

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

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s − 0.380·3-s + 0.500·4-s − 0.253·5-s − 0.269·6-s + 0.397·7-s + 0.353·8-s − 0.855·9-s − 0.179·10-s + 1.20·11-s − 0.190·12-s − 0.948·13-s + 0.281·14-s + 0.0965·15-s + 0.250·16-s + 0.472·17-s − 0.604·18-s − 0.231·19-s − 0.126·20-s − 0.151·21-s + 0.849·22-s + 0.398·23-s − 0.134·24-s − 0.935·25-s − 0.670·26-s + 0.705·27-s + 0.198·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 2 ^{s/2} \, \Gamma_{\R}(s+(1 + 5.41i)) \, \Gamma_{\R}(s+(1 - 5.41i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 2,\ (1 + 5.41733480684i, 1 - 5.41733480684i:\ ),\ 1)$

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