# Properties

 Degree 4 Conductor $1$ Sign $1$ Primitive yes Self-dual yes Analytic rank 0

## Dirichlet series

 $L(s,f)$  = 1 + 2.06·2-s + 1.79·3-s + 1.79·4-s + 0.454·5-s + 3.71·6-s + 0.0379·7-s + 0.675·8-s + 0.779·9-s + 0.939·10-s − 0.992·11-s + 3.23·12-s − 0.332·13-s + 0.0785·14-s + 0.816·15-s + 0.219·16-s − 0.298·17-s + 1.61·18-s + 0.400·19-s + 0.817·20-s + 0.0682·21-s − 2.05·22-s + 0.698·23-s + 1.21·24-s + 1.03·25-s − 0.687·26-s − 1.21·27-s + 0.0682·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+19.8i) \, \Gamma_{\R}(s+8.53i) \, \Gamma_{\R}(s-19.8i) \, \Gamma_{\R}(s-8.53i) \, L(s,f)\cr =\mathstrut & \,\Lambda(1-s,f) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(4,\ 1,\ (19.81936708928i, 8.53107821814i, -19.81936708928i, -8.53107821814i:\ ),\ 1)$

## Euler product

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

## 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