# Properties

 Degree 4 Conductor $229^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s − 4-s + 6·5-s − 3·9-s + 6·11-s − 2·12-s + 12·15-s − 3·16-s − 6·17-s − 2·19-s − 6·20-s + 17·25-s − 14·27-s + 12·33-s + 3·36-s + 4·37-s − 2·43-s − 6·44-s − 18·45-s − 6·48-s + 14·49-s − 12·51-s + 12·53-s + 36·55-s − 4·57-s − 12·60-s + 10·61-s + ⋯
 L(s)  = 1 + 1.15·3-s − 1/2·4-s + 2.68·5-s − 9-s + 1.80·11-s − 0.577·12-s + 3.09·15-s − 3/4·16-s − 1.45·17-s − 0.458·19-s − 1.34·20-s + 17/5·25-s − 2.69·27-s + 2.08·33-s + 1/2·36-s + 0.657·37-s − 0.304·43-s − 0.904·44-s − 2.68·45-s − 0.866·48-s + 2·49-s − 1.68·51-s + 1.64·53-s + 4.85·55-s − 0.529·57-s − 1.54·60-s + 1.28·61-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$52441$$    =    $$229^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{229} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 52441,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $2.54958$ $L(\frac12)$ $\approx$ $2.54958$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 229$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 229$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad229$C_2$ $$1 - 14 T + p T^{2}$$
good2$V_4$ $$1 + T^{2} + p^{2} T^{4}$$
3$C_2$ $$( 1 - T + p T^{2} )^{2}$$
5$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
7$C_2$ $$( 1 - p T^{2} )^{2}$$
11$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - p T^{2} )^{2}$$
17$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + T + p T^{2} )^{2}$$
23$V_4$ $$1 - 26 T^{2} + p^{2} T^{4}$$
29$V_4$ $$1 - 38 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - p T^{2} )^{2}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
43$C_2$ $$( 1 + T + p T^{2} )^{2}$$
47$V_4$ $$1 - 14 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
59$V_4$ $$1 - 38 T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
67$V_4$ $$1 + 46 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 + 15 T + p T^{2} )^{2}$$
73$V_4$ $$1 + 34 T^{2} + p^{2} T^{4}$$
79$V_4$ $$1 + 22 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 + 9 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}