# Properties

 Degree 4 Conductor $3^{2} \cdot 47^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 2

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 4·5-s − 6·7-s + 4·8-s + 9-s + 8·10-s − 8·11-s − 2·13-s + 12·14-s − 4·16-s + 2·17-s − 2·18-s − 12·19-s + 16·22-s + 12·23-s + 3·25-s + 4·26-s + 2·31-s − 4·34-s + 24·35-s + 2·37-s + 24·38-s − 16·40-s − 4·41-s − 6·43-s − 4·45-s − 24·46-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.78·5-s − 2.26·7-s + 1.41·8-s + 1/3·9-s + 2.52·10-s − 2.41·11-s − 0.554·13-s + 3.20·14-s − 16-s + 0.485·17-s − 0.471·18-s − 2.75·19-s + 3.41·22-s + 2.50·23-s + 3/5·25-s + 0.784·26-s + 0.359·31-s − 0.685·34-s + 4.05·35-s + 0.328·37-s + 3.89·38-s − 2.52·40-s − 0.624·41-s − 0.914·43-s − 0.596·45-s − 3.53·46-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 19881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 19881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$19881$$    =    $$3^{2} \cdot 47^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{19881} (1, \cdot )$ Sato-Tate : $G_{3,3}$ primitive : no self-dual : yes analytic rank = 2 Selberg data = $(4,\ 19881,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;47\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
47$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good2$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + p T + p T^{2} )$$
5$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} )$$
7$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
11$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
23$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} )$$
29$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
37$C_2$ $$( 1 - T + p T^{2} )^{2}$$
41$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} )$$
83$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 18 T + p T^{2} )$$
89$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} )$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−16.3084950271, −15.7159115639, −15.5159935893, −15.1680730202, −14.6489059291, −13.3552681657, −13.2982222373, −12.78597999, −12.6657932749, −11.9075398839, −11.064550122, −10.5981068261, −10.1950027975, −9.89655043622, −9.22229106146, −8.53725475056, −8.40526764581, −7.63249491007, −7.30736849694, −6.71008245044, −5.83486052932, −4.7534983042, −4.36466935125, −3.31847810115, −2.77908676833, 0, 0, 2.77908676833, 3.31847810115, 4.36466935125, 4.7534983042, 5.83486052932, 6.71008245044, 7.30736849694, 7.63249491007, 8.40526764581, 8.53725475056, 9.22229106146, 9.89655043622, 10.1950027975, 10.5981068261, 11.064550122, 11.9075398839, 12.6657932749, 12.78597999, 13.2982222373, 13.3552681657, 14.6489059291, 15.1680730202, 15.5159935893, 15.7159115639, 16.3084950271