# Properties

 Label 4-933397-1.1-c1e2-0-0 Degree $4$ Conductor $933397$ Sign $1$ Analytic cond. $59.5142$ Root an. cond. $2.77750$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s − 4-s − 6-s + 2·7-s + 8-s − 9-s + 11-s − 12-s + 5·13-s − 2·14-s − 16-s + 6·17-s + 18-s − 2·19-s + 2·21-s − 22-s − 4·23-s + 24-s − 6·25-s − 5·26-s − 2·28-s + 2·29-s + 7·31-s + 5·32-s + 33-s − 6·34-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s − 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.38·13-s − 0.534·14-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.436·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 6/5·25-s − 0.980·26-s − 0.377·28-s + 0.371·29-s + 1.25·31-s + 0.883·32-s + 0.174·33-s − 1.02·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 933397 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 933397 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$933397$$ Sign: $1$ Analytic conductor: $$59.5142$$ Root analytic conductor: $$2.77750$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{933397} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 933397,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.411263916$$ $$L(\frac12)$$ $$\approx$$ $$1.411263916$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad933397$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 526 T + p T^{2} )$$
good2$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + p T + p T^{2} )$$
3$D_{4}$ $$1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$D_{4}$ $$1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$D_{4}$ $$1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
19$C_4$ $$1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$D_{4}$ $$1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + p T^{2} )$$
53$D_{4}$ $$1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
71$D_{4}$ $$1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 18 T + 174 T^{2} - 18 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 16 T + 182 T^{2} + 16 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 2 T + 154 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$