# Properties

 Label 4-1783296-1.1-c1e2-0-7 Degree $4$ Conductor $1783296$ Sign $-1$ Analytic cond. $113.704$ Root an. cond. $3.26546$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s + 9-s − 3·11-s + 2·17-s − 8·19-s + 6·25-s + 27-s − 3·33-s + 7·41-s − 9·43-s − 5·49-s + 2·51-s − 8·57-s − 23·59-s + 7·67-s + 13·73-s + 6·75-s + 81-s + 19·83-s − 3·89-s − 2·97-s − 3·99-s + 7·107-s − 5·113-s − 3·121-s + 7·123-s + 127-s + ⋯
 L(s)  = 1 + 0.577·3-s + 1/3·9-s − 0.904·11-s + 0.485·17-s − 1.83·19-s + 6/5·25-s + 0.192·27-s − 0.522·33-s + 1.09·41-s − 1.37·43-s − 5/7·49-s + 0.280·51-s − 1.05·57-s − 2.99·59-s + 0.855·67-s + 1.52·73-s + 0.692·75-s + 1/9·81-s + 2.08·83-s − 0.317·89-s − 0.203·97-s − 0.301·99-s + 0.676·107-s − 0.470·113-s − 0.272·121-s + 0.631·123-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1783296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1783296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1783296$$    =    $$2^{9} \cdot 3^{4} \cdot 43$$ Sign: $-1$ Analytic conductor: $$113.704$$ Root analytic conductor: $$3.26546$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 1783296,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_1$ $$1 - T$$
43$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 8 T + p T^{2} )$$
good5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
11$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
13$C_2^2$ $$1 + 9 T^{2} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
19$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
23$C_2^2$ $$1 + 32 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 - 20 T^{2} + p^{2} T^{4}$$
41$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
47$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 19 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 + 10 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
61$C_2^2$ $$1 + 112 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
71$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 - 6 T + p T^{2} )$$
79$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 - 9 T + p T^{2} )$$
89$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 2 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$