# Properties

 Degree 4 Conductor $11^{2} \cdot 89^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 8·4-s + 2·5-s − 8·8-s − 5·9-s − 8·10-s + 2·11-s − 4·16-s − 4·17-s + 20·18-s + 16·20-s − 8·22-s − 7·25-s + 32·32-s + 16·34-s − 40·36-s − 16·40-s + 16·44-s − 10·45-s + 16·47-s − 10·49-s + 28·50-s − 12·53-s + 4·55-s − 64·64-s − 14·67-s − 32·68-s + ⋯
 L(s)  = 1 − 2.82·2-s + 4·4-s + 0.894·5-s − 2.82·8-s − 5/3·9-s − 2.52·10-s + 0.603·11-s − 16-s − 0.970·17-s + 4.71·18-s + 3.57·20-s − 1.70·22-s − 7/5·25-s + 5.65·32-s + 2.74·34-s − 6.66·36-s − 2.52·40-s + 2.41·44-s − 1.49·45-s + 2.33·47-s − 1.42·49-s + 3.95·50-s − 1.64·53-s + 0.539·55-s − 8·64-s − 1.71·67-s − 3.88·68-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 958441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 958441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$958441$$    =    $$11^{2} \cdot 89^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{958441} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 958441,\ (\ :1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.1707542378$$ $$L(\frac12)$$ $$\approx$$ $$0.1707542378$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{11,\;89\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{11,\;89\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad11$C_1$ $$( 1 - T )^{2}$$
89$C_2$ $$1 - 15 T + p T^{2}$$
good2$C_2$ $$( 1 + p T + p T^{2} )^{2}$$
3$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
5$C_2$ $$( 1 - T + p T^{2} )^{2}$$
7$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
37$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
41$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
43$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
61$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
67$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
\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}