Properties

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

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Dirichlet series

$L(s,f)$  = 1  − 0.707·2-s − 0.949·3-s + 0.499·4-s − 0.869·5-s + 0.671·6-s − 0.0613·7-s − 0.353·8-s − 0.0987·9-s + 0.614·10-s − 0.0741·11-s − 0.474·12-s − 0.0863·13-s + 0.0433·14-s + 0.825·15-s + 0.250·16-s − 1.69·17-s + 0.0698·18-s + 1.42·19-s − 0.434·20-s + 0.0582·21-s + 0.0524·22-s − 0.759·23-s + 0.335·24-s − 0.243·25-s + 0.0610·26-s + 1.04·27-s − 0.0306·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 2 ^{s/2} \, \Gamma_{\R}(s+(1 + 7.22i)) \, \Gamma_{\R}(s+(1 - 7.22i)) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 2,\ (1 + 7.22087197596i, 1 - 7.22087197596i:\ ),\ -1)$

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