# Properties

 Label 4-115200-1.1-c1e2-0-30 Degree $4$ Conductor $115200$ Sign $-1$ Analytic cond. $7.34525$ Root an. cond. $1.64627$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s + 3·9-s − 8·11-s − 12·17-s − 8·19-s + 25-s + 4·27-s − 16·33-s − 12·41-s + 24·43-s − 14·49-s − 24·51-s − 16·57-s + 24·59-s + 8·67-s − 12·73-s + 2·75-s + 5·81-s − 24·83-s + 20·89-s + 4·97-s − 24·99-s − 8·107-s − 12·113-s + 26·121-s − 24·123-s + 127-s + ⋯
 L(s)  = 1 + 1.15·3-s + 9-s − 2.41·11-s − 2.91·17-s − 1.83·19-s + 1/5·25-s + 0.769·27-s − 2.78·33-s − 1.87·41-s + 3.65·43-s − 2·49-s − 3.36·51-s − 2.11·57-s + 3.12·59-s + 0.977·67-s − 1.40·73-s + 0.230·75-s + 5/9·81-s − 2.63·83-s + 2.11·89-s + 0.406·97-s − 2.41·99-s − 0.773·107-s − 1.12·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$115200$$    =    $$2^{9} \cdot 3^{2} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$7.34525$$ Root analytic conductor: $$1.64627$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 115200,\ (\ :1/2, 1/2),\ -1)$$

## 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 $$1$$
3$C_1$ $$( 1 - T )^{2}$$
5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
31$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$