# Properties

 Degree 4 Conductor $2^{4} \cdot 5^{2} \cdot 13 \cdot 17$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 4-s + 2·5-s + 3·8-s − 6·9-s − 2·10-s − 13-s − 16-s + 3·17-s + 6·18-s − 2·20-s + 3·25-s + 26-s − 4·29-s − 5·32-s − 3·34-s + 6·36-s − 4·37-s + 6·40-s − 12·41-s − 12·45-s + 2·49-s − 3·50-s + 52-s − 20·53-s + 4·58-s + 12·61-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 2·9-s − 0.632·10-s − 0.277·13-s − 1/4·16-s + 0.727·17-s + 1.41·18-s − 0.447·20-s + 3/5·25-s + 0.196·26-s − 0.742·29-s − 0.883·32-s − 0.514·34-s + 36-s − 0.657·37-s + 0.948·40-s − 1.87·41-s − 1.78·45-s + 2/7·49-s − 0.424·50-s + 0.138·52-s − 2.74·53-s + 0.525·58-s + 1.53·61-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 88400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 88400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$88400$$    =    $$2^{4} \cdot 5^{2} \cdot 13 \cdot 17$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{88400} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(4,\ 88400,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;13,\;17\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;13,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 + T + p T^{2}$$
5$C_1$ $$( 1 - T )^{2}$$
13$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 2 T + p T^{2} )$$
17$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 2 T + p T^{2} )$$
good3$C_2$ $$( 1 + p T^{2} )^{2}$$
7$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$V_4$ $$1 - 10 T^{2} + p^{2} T^{4}$$
19$V_4$ $$1 - 10 T^{2} + p^{2} T^{4}$$
23$V_4$ $$1 - 34 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
31$V_4$ $$1 - 50 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
43$V_4$ $$1 + 54 T^{2} + p^{2} T^{4}$$
47$V_4$ $$1 - 34 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
59$V_4$ $$1 + 70 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$V_4$ $$1 - 106 T^{2} + p^{2} T^{4}$$
71$V_4$ $$1 - 34 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
83$V_4$ $$1 + 54 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}