# Properties

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

## 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

$$$L(s,f) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$$

## Imaginary part of the first few zeros on the critical line

−24.223274940, −22.660338142, −21.022531596, −14.944466160, −14.004231120, −13.074038138, −5.102901815, −3.371715172, −2.480449521, 2.480449521, 3.371715172, 5.102901815, 13.074038138, 14.004231120, 14.944466160, 21.022531596, 22.660338142, 24.223274940