Properties

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

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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 + ⋯

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

\( d \)  =  \(2\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 1,\ (1 + 12.1730083247i, 1 - 12.1730083247i:\ ),\ -1)$

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

Zeros not available.

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

Plot not available.