# Properties

 Label 4-332928-1.1-c1e2-0-4 Degree $4$ Conductor $332928$ Sign $1$ Analytic cond. $21.2277$ Root an. cond. $2.14647$ 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 − 3·5-s + 6-s + 8-s − 2·9-s − 3·10-s + 12-s − 3·15-s + 16-s − 2·18-s − 2·19-s − 3·20-s + 6·23-s + 24-s − 25-s − 5·27-s + 12·29-s − 3·30-s + 32-s − 2·36-s − 2·38-s − 3·40-s + 7·43-s + 6·45-s + 6·46-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.288·12-s − 0.774·15-s + 1/4·16-s − 0.471·18-s − 0.458·19-s − 0.670·20-s + 1.25·23-s + 0.204·24-s − 1/5·25-s − 0.962·27-s + 2.22·29-s − 0.547·30-s + 0.176·32-s − 1/3·36-s − 0.324·38-s − 0.474·40-s + 1.06·43-s + 0.894·45-s + 0.884·46-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$332928$$    =    $$2^{7} \cdot 3^{2} \cdot 17^{2}$$ Sign: $1$ Analytic conductor: $$21.2277$$ Root analytic conductor: $$2.14647$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 332928,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.323858007$$ $$L(\frac12)$$ $$\approx$$ $$2.323858007$$ $$L(\frac{3}{2})$$ 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)$
bad2$C_1$ $$1 - T$$
3$C_2$ $$1 - T + p T^{2}$$
17$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good5$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 3 T + p T^{2} )$$
7$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
13$C_2^2$ $$1 - 20 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 + T + p T^{2} )^{2}$$
23$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + p T^{2} )$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
31$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 28 T^{2} + p^{2} T^{4}$$
41$C_2^2$ $$1 - 35 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
59$C_2^2$ $$1 + T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 80 T^{2} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
71$C_2$$\times$$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
79$C_2^2$ $$1 - 32 T^{2} + p^{2} T^{4}$$
83$C_2^2$ $$1 - 113 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 61 T^{2} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$