# Properties

 Degree 4 Conductor $2^{2} \cdot 3^{6} \cdot 7$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 4-s + 13-s + 16-s + 4·19-s + 8·25-s − 8·31-s − 5·37-s + 7·43-s − 6·49-s + 52-s + 7·61-s + 64-s − 17·67-s + 13·73-s + 4·76-s − 11·79-s − 20·97-s + 8·100-s − 8·103-s − 14·109-s − 4·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + ⋯
 L(s)  = 1 + 1/2·4-s + 0.277·13-s + 1/4·16-s + 0.917·19-s + 8/5·25-s − 1.43·31-s − 0.821·37-s + 1.06·43-s − 6/7·49-s + 0.138·52-s + 0.896·61-s + 1/8·64-s − 2.07·67-s + 1.52·73-s + 0.458·76-s − 1.23·79-s − 2.03·97-s + 4/5·100-s − 0.788·103-s − 1.34·109-s − 0.363·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.410·148-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20412 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20412 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$20412$$    =    $$2^{2} \cdot 3^{6} \cdot 7$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{20412} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 20412,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $1.321137289$ $L(\frac12)$ $\approx$ $1.321137289$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
3 $$1$$
7$C_1$$\times$$C_2$ $$( 1 - T )( 1 + T + p T^{2} )$$
good5$V_4$ $$1 - 8 T^{2} + p^{2} T^{4}$$
11$V_4$ $$1 + 4 T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$V_4$ $$1 - 20 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
23$V_4$ $$1 - 26 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
31$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
37$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
41$V_4$ $$1 + 28 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )$$
47$V_4$ $$1 + 31 T^{2} + p^{2} T^{4}$$
53$V_4$ $$1 - 29 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
71$V_4$ $$1 - 20 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
89$V_4$ $$1 + 16 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}