# Properties

 Degree 4 Conductor $3^{4} \cdot 89^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 4·4-s + 12·16-s − 10·25-s + 14·49-s − 32·64-s − 2·67-s + 10·73-s − 14·79-s + 22·97-s + 40·100-s − 26·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 2·4-s + 3·16-s − 2·25-s + 2·49-s − 4·64-s − 0.244·67-s + 1.17·73-s − 1.57·79-s + 2.23·97-s + 4·100-s − 2.49·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 641601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 641601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$641601$$    =    $$3^{4} \cdot 89^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{641601} (1, \cdot )$ Sato-Tate : $N(\mathrm{U}(1))$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 641601,\ (\ :1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.7310814825$$ $$L(\frac12)$$ $$\approx$$ $$0.7310814825$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;89\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;89\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 $$1$$
89$C_2$ $$1 + p T^{2}$$
good2$C_2$ $$( 1 + p T^{2} )^{2}$$
5$C_2$ $$( 1 + p T^{2} )^{2}$$
7$C_2$ $$( 1 - p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - p T^{2} )^{2}$$
17$C_2$ $$( 1 + p T^{2} )^{2}$$
19$C_2$ $$( 1 - p T^{2} )^{2}$$
23$C_2^2$ $$1 + 43 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 31 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - p T^{2} )^{2}$$
37$C_2$ $$( 1 - p T^{2} )^{2}$$
41$C_2^2$ $$1 + 7 T^{2} + p^{2} T^{4}$$
43$C_2$ $$( 1 - p T^{2} )^{2}$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 + p T^{2} )^{2}$$
59$C_2^2$ $$1 - 29 T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 - p T^{2} )^{2}$$
67$C_2$ $$( 1 + T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
83$C_2^2$ $$1 - 77 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 11 T + p T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}