# Properties

 Degree 4 Conductor $2^{2} \cdot 11^{2} \cdot 19^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s + 3·4-s − 4·5-s − 4·8-s + 6·9-s + 8·10-s + 2·11-s − 12·13-s + 5·16-s − 12·18-s − 6·19-s − 12·20-s − 4·22-s − 8·23-s + 2·25-s + 24·26-s + 12·29-s − 6·32-s + 18·36-s + 12·38-s + 16·40-s + 6·44-s − 24·45-s + 16·46-s − 16·47-s + 4·49-s − 4·50-s + ⋯
 L(s)  = 1 − 1.41·2-s + 3/2·4-s − 1.78·5-s − 1.41·8-s + 2·9-s + 2.52·10-s + 0.603·11-s − 3.32·13-s + 5/4·16-s − 2.82·18-s − 1.37·19-s − 2.68·20-s − 0.852·22-s − 1.66·23-s + 2/5·25-s + 4.70·26-s + 2.22·29-s − 1.06·32-s + 3·36-s + 1.94·38-s + 2.52·40-s + 0.904·44-s − 3.57·45-s + 2.35·46-s − 2.33·47-s + 4/7·49-s − 0.565·50-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 : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 174724,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.2976551487$ $L(\frac12)$ $\approx$ $0.2976551487$ $L(\frac{3}{2})$ 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$C_1$ $$( 1 + T )^{2}$$
11$C_2$ $$1 - 2 T + p T^{2}$$
19$C_2$ $$1 + 6 T + p T^{2}$$
good3$C_2$ $$( 1 - p T^{2} )^{2}$$
5$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
7$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
17$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
23$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
31$C_2^2$ $$1 + 28 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 16 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 - p T^{2} )^{2}$$
47$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
53$C_2^2$ $$1 - 16 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - p T^{2} )^{2}$$
61$C_2^2$ $$1 - 112 T^{2} + p^{2} T^{4}$$
67$C_2$ $$( 1 - p T^{2} )^{2}$$
71$C_2^2$ $$1 - 52 T^{2} + p^{2} T^{4}$$
73$C_2$ $$( 1 - p T^{2} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2^2$ $$1 - 126 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - p T^{2} )^{2}$$
97$C_2^2$ $$1 + 166 T^{2} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}