# Properties

 Degree 4 Conductor $3^{4} \cdot 19^{3}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 3·4-s + 4·7-s + 6·13-s + 5·16-s + 19-s + 8·25-s + 12·28-s − 10·37-s + 2·49-s + 18·52-s + 16·61-s + 3·64-s − 12·67-s + 4·73-s + 3·76-s − 12·79-s + 24·91-s + 6·97-s + 24·100-s − 4·103-s − 18·109-s + 20·112-s − 4·121-s + 127-s + 131-s + 4·133-s + 137-s + ⋯
 L(s)  = 1 + 3/2·4-s + 1.51·7-s + 1.66·13-s + 5/4·16-s + 0.229·19-s + 8/5·25-s + 2.26·28-s − 1.64·37-s + 2/7·49-s + 2.49·52-s + 2.04·61-s + 3/8·64-s − 1.46·67-s + 0.468·73-s + 0.344·76-s − 1.35·79-s + 2.51·91-s + 0.609·97-s + 12/5·100-s − 0.394·103-s − 1.72·109-s + 1.88·112-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$555579$$    =    $$3^{4} \cdot 19^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{555579} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 555579,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $4.139988395$ $L(\frac12)$ $\approx$ $4.139988395$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;19\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 $$1$$
19$C_1$ $$1 - T$$
good2$C_2^2$ $$1 - 3 T^{2} + p^{2} T^{4}$$
5$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
11$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
17$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
23$C_2^2$ $$1 - 36 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 86 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
71$C_2^2$ $$1 + 98 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2^2$ $$1 + 116 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 94 T^{2} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}