# Properties

 Label 4-10976-1.1-c1e2-0-0 Degree $4$ Conductor $10976$ Sign $1$ Analytic cond. $0.699839$ Root an. cond. $0.914638$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s − 4-s + 7-s − 3·8-s + 2·9-s + 4·11-s + 14-s − 16-s + 2·18-s + 4·22-s + 4·23-s + 6·25-s − 28-s − 16·29-s + 5·32-s − 2·36-s − 8·37-s − 12·43-s − 4·44-s + 4·46-s + 49-s + 6·50-s − 4·53-s − 3·56-s − 16·58-s + 2·63-s + 7·64-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 2/3·9-s + 1.20·11-s + 0.267·14-s − 1/4·16-s + 0.471·18-s + 0.852·22-s + 0.834·23-s + 6/5·25-s − 0.188·28-s − 2.97·29-s + 0.883·32-s − 1/3·36-s − 1.31·37-s − 1.82·43-s − 0.603·44-s + 0.589·46-s + 1/7·49-s + 0.848·50-s − 0.549·53-s − 0.400·56-s − 2.10·58-s + 0.251·63-s + 7/8·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$10976$$    =    $$2^{5} \cdot 7^{3}$$ Sign: $1$ Analytic conductor: $$0.699839$$ Root analytic conductor: $$0.914638$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{10976} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 10976,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.312347190$$ $$L(\frac12)$$ $$\approx$$ $$1.312347190$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 - T + p T^{2}$$
7$C_1$ $$1 - T$$
good3$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
13$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
31$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
59$C_2^2$ $$1 + 78 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 38 T^{2} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2^2$ $$1 - 114 T^{2} + p^{2} T^{4}$$
79$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 110 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 - 170 T^{2} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$