# Properties

 Degree 4 Conductor $2^{2} \cdot 11^{2} \cdot 19^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 + 4-s − 4·5-s − 3·9-s + 2·11-s + 16-s − 4·20-s + 10·23-s + 2·25-s − 3·36-s + 2·44-s + 12·45-s − 16·47-s − 5·49-s − 8·55-s + 64-s − 4·80-s + 10·92-s − 6·99-s + 2·100-s − 40·115-s − 7·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + ⋯
 L(s)  = 1 + 1/2·4-s − 1.78·5-s − 9-s + 0.603·11-s + 1/4·16-s − 0.894·20-s + 2.08·23-s + 2/5·25-s − 1/2·36-s + 0.301·44-s + 1.78·45-s − 2.33·47-s − 5/7·49-s − 1.07·55-s + 1/8·64-s − 0.447·80-s + 1.04·92-s − 0.603·99-s + 1/5·100-s − 3.73·115-s − 0.636·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$174724$$    =    $$2^{2} \cdot 11^{2} \cdot 19^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{174724} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(4,\ 174724,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;11,\;19\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
11$C_2$ $$1 - 2 T + p T^{2}$$
19$C_2$ $$1 + p T^{2}$$
good3$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
7$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
17$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
23$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
31$C_2^2$ $$1 - 26 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 - 38 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
43$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
47$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
53$C_2^2$ $$1 - 97 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 109 T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 + p T^{2} )^{2}$$
67$C_2^2$ $$1 + 91 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 - p T^{2} )^{2}$$
73$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
79$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
83$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
89$C_2^2$ $$1 - 142 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}