Properties

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

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Dirichlet series

$L(s,f)$  = 1  + 1.549·2-s + 0.246·3-s + 1.400·4-s + 0.737·5-s + 0.382·6-s − 0.261·7-s + 0.620·8-s − 0.939·9-s + 1.141·10-s − 0.953·11-s + 0.345·12-s + 0.278·13-s − 0.405·14-s + 0.181·15-s − 0.439·16-s + 1.307·17-s − 1.454·18-s + 0.092·19-s + 1.032·20-s − 0.064·21-s − 1.477·22-s + 1.138·23-s + 0.153·24-s − 0.456·25-s + 0.431·26-s − 0.478·27-s − 0.366·28-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+13.77i)\Gamma_{\R}(s-13.77 i) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 1,\ (13.7797513518907i, -13.7797513518907i:\ ),\ 1)$

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