Properties

 Label 4-84e2-1.1-c1e2-0-4 Degree $4$ Conductor $7056$ Sign $-1$ Analytic cond. $0.449896$ Root an. cond. $0.818989$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

Origins of factors

Dirichlet series

 L(s)  = 1 − 2-s − 4-s − 4·5-s + 3·8-s + 9-s + 4·10-s − 4·13-s − 16-s − 12·17-s − 18-s + 4·20-s + 2·25-s + 4·26-s − 4·29-s − 5·32-s + 12·34-s − 36-s + 12·37-s − 12·40-s + 4·41-s − 4·45-s + 49-s − 2·50-s + 4·52-s + 12·53-s + 4·58-s − 4·61-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s + 1/3·9-s + 1.26·10-s − 1.10·13-s − 1/4·16-s − 2.91·17-s − 0.235·18-s + 0.894·20-s + 2/5·25-s + 0.784·26-s − 0.742·29-s − 0.883·32-s + 2.05·34-s − 1/6·36-s + 1.97·37-s − 1.89·40-s + 0.624·41-s − 0.596·45-s + 1/7·49-s − 0.282·50-s + 0.554·52-s + 1.64·53-s + 0.525·58-s − 0.512·61-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$7056$$    =    $$2^{4} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$0.449896$$ Root analytic conductor: $$0.818989$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{7056} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 7056,\ (\ :1/2, 1/2),\ -1)$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 + T + p T^{2}$$
3$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good5$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2$ $$( 1 + 14 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 18 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$