# Properties

 Degree 4 Conductor $2^{2} \cdot 3 \cdot 5^{2} \cdot 17^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s − 4-s − 2·5-s − 2·7-s + 12-s + 2·15-s + 16-s + 2·17-s + 8·19-s + 2·20-s + 2·21-s − 2·23-s − 25-s + 4·27-s + 2·28-s + 4·35-s − 8·37-s − 48-s − 2·49-s − 2·51-s − 8·57-s − 4·59-s − 2·60-s − 64-s − 2·68-s + 2·69-s − 8·73-s + ⋯
 L(s)  = 1 − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.755·7-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 1.83·19-s + 0.447·20-s + 0.436·21-s − 0.417·23-s − 1/5·25-s + 0.769·27-s + 0.377·28-s + 0.676·35-s − 1.31·37-s − 0.144·48-s − 2/7·49-s − 0.280·51-s − 1.05·57-s − 0.520·59-s − 0.258·60-s − 1/8·64-s − 0.242·68-s + 0.240·69-s − 0.936·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$86700$$    =    $$2^{2} \cdot 3 \cdot 5^{2} \cdot 17^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{86700} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(4,\ 86700,\ (\ :1/2, 1/2),\ -1)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;17\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + T^{2}$$
3$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 2 T + p T^{2} )$$
5$C_2$ $$1 + 2 T + p T^{2}$$
17$C_2$ $$1 - 2 T + p T^{2}$$
good7$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
13$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
23$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
29$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
37$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2^2$ $$1 + 18 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 54 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
61$C_2^2$ $$1 + 38 T^{2} + p^{2} T^{4}$$
67$C_2^2$ $$1 - 106 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
73$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
79$C_2^2$ $$1 - 70 T^{2} + p^{2} T^{4}$$
83$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
97$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}