Properties

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

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Dirichlet series

$L(s,f)$  = 1  + 0.289·2-s − 1.201·3-s − 0.916·4-s + 0.039·5-s − 0.347·6-s + 0.448·7-s − 0.554·8-s + 0.444·9-s + 0.011·10-s − 0.691·11-s + 1.101·12-s − 0.802·13-s + 0.129·14-s − 0.047·15-s + 0.756·16-s − 1.037·17-s + 0.128·18-s + 0.637·19-s − 0.036·20-s − 0.538·21-s − 0.200·22-s + 0.508·23-s + 0.666·24-s − 0.998·25-s − 0.232·26-s + 0.667·27-s − 0.410·28-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+(1 + 12.17 i) )\Gamma_{\R}(s+(1-12.17 i) ) \cdot L(s,f)\cr =\mathstrut & -\Lambda(1-s,f) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 1,\ (1 + 12.1730083246797i, 1-12.1730083246797i:\ ),\ -1)$

Euler product

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

Imaginary part of the first few zeros on the critical line

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