Properties

Degree 4
Conductor 1
Sign $1$
Self-dual yes

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Dirichlet series

$L(s,f)$  = 1  + 2.066·2-s + 1.797·3-s + 1.799·4-s + 0.454·5-s + 3.715·6-s + 0.037·7-s + 0.675·8-s + 0.779·9-s + 0.939·10-s − 0.992·11-s + 3.235·12-s − 0.332·13-s + 0.078·14-s + 0.816·15-s + 0.219·16-s − 0.298·17-s + 1.610·18-s + 0.400·19-s + 0.817·20-s + 0.068·21-s − 2.051·22-s + 0.698·23-s + 1.214·24-s + 1.034·25-s − 0.687·26-s − 1.210·27-s + 0.068·28-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+19.81i)\Gamma_{\R}(s+8.531i)\Gamma_{\R}(s-19.81 i)\Gamma_{\R}(s-8.531 i) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 1,\ (19.8193670893i, 8.53107821814i, -19.8193670893i, -8.53107821814i:\ ),\ 1)$

Euler product

\[\begin{equation} L(s,f) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

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