# Properties

 Degree 4 Conductor $2^{2} \cdot 3^{2} \cdot 31^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s − 3·9-s + 4·13-s + 16-s + 8·19-s − 6·25-s − 2·31-s − 3·36-s + 20·37-s + 16·43-s − 14·49-s + 4·52-s − 12·61-s + 64-s − 24·67-s + 20·73-s + 8·76-s − 16·79-s + 9·81-s + 4·97-s − 6·100-s + 16·103-s − 4·109-s − 12·117-s − 22·121-s − 2·124-s + 127-s + ⋯
 L(s)  = 1 + 1/2·4-s − 9-s + 1.10·13-s + 1/4·16-s + 1.83·19-s − 6/5·25-s − 0.359·31-s − 1/2·36-s + 3.28·37-s + 2.43·43-s − 2·49-s + 0.554·52-s − 1.53·61-s + 1/8·64-s − 2.93·67-s + 2.34·73-s + 0.917·76-s − 1.80·79-s + 81-s + 0.406·97-s − 3/5·100-s + 1.57·103-s − 0.383·109-s − 1.10·117-s − 2·121-s − 0.179·124-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$34596$$    =    $$2^{2} \cdot 3^{2} \cdot 31^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{34596} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 34596,\ (\ :1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$1.433324264$$ $$L(\frac12)$$ $$\approx$$ $$1.433324264$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;31\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
3$C_2$ $$1 + p T^{2}$$
31$C_1$ $$( 1 + T )^{2}$$
good5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
37$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
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$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
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$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
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}