# Properties

 Degree 4 Conductor $2^{7} \cdot 31^{2}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s + 8-s − 6·9-s + 16-s − 12·17-s − 6·18-s + 16·23-s − 6·25-s − 2·31-s + 32-s − 12·34-s − 6·36-s − 12·41-s + 16·46-s − 16·47-s − 14·49-s − 6·50-s − 2·62-s + 64-s − 12·68-s + 16·71-s − 6·72-s + 20·73-s − 16·79-s + 27·81-s − 12·82-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s + 1/4·16-s − 2.91·17-s − 1.41·18-s + 3.33·23-s − 6/5·25-s − 0.359·31-s + 0.176·32-s − 2.05·34-s − 36-s − 1.87·41-s + 2.35·46-s − 2.33·47-s − 2·49-s − 0.848·50-s − 0.254·62-s + 1/8·64-s − 1.45·68-s + 1.89·71-s − 0.707·72-s + 2.34·73-s − 1.80·79-s + 3·81-s − 1.32·82-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$123008$$    =    $$2^{7} \cdot 31^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{123008} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 1 Selberg data = $(4,\ 123008,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;31\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$1 - T$$
31$C_1$ $$( 1 + T )^{2}$$
good3$C_2$ $$( 1 + p T^{2} )^{2}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
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}