# Properties

 Degree 4 Conductor $2^{4} \cdot 11^{2} \cdot 19^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 2·4-s − 6·5-s + 5·9-s + 4·16-s + 12·20-s + 17·25-s − 10·36-s − 30·45-s − 14·49-s − 8·64-s − 24·80-s + 16·81-s − 34·100-s − 11·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 20·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + ⋯
 L(s)  = 1 − 4-s − 2.68·5-s + 5/3·9-s + 16-s + 2.68·20-s + 17/5·25-s − 5/3·36-s − 4.47·45-s − 2·49-s − 64-s − 2.68·80-s + 16/9·81-s − 3.39·100-s − 121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/3·144-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 + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$698896$$    =    $$2^{4} \cdot 11^{2} \cdot 19^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{698896} (1, \cdot )$ Sato-Tate : $N(\mathrm{U}(1))$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(4,\ 698896,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;11,\;19\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + p T^{2}$$
11$C_2$ $$1 + p T^{2}$$
19$C_2$ $$1 + p T^{2}$$
good3$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 + p T^{2} )^{2}$$
17$C_2$ $$( 1 - p T^{2} )^{2}$$
23$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2^2$ $$1 - 37 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2^2$ $$1 + 107 T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 - p T^{2} )^{2}$$
67$C_2^2$ $$1 + 35 T^{2} + p^{2} T^{4}$$
71$C_2^2$ $$1 - 133 T^{2} + p^{2} T^{4}$$
73$C_2$ $$( 1 - p T^{2} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
97$C_2$ $$( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}