# Properties

 Degree 4 Conductor $11^{3} \cdot 19^{2}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

# 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 + 11-s + 8·13-s − 4·16-s + 20·18-s + 16·20-s − 4·22-s − 2·23-s − 7·25-s − 32·26-s + 32·32-s − 40·36-s − 16·40-s − 16·41-s + 8·44-s − 10·45-s + 8·46-s + 16·47-s − 10·49-s + 28·50-s + 64·52-s + 2·55-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 + 0.301·11-s + 2.21·13-s − 16-s + 4.71·18-s + 3.57·20-s − 0.852·22-s − 0.417·23-s − 7/5·25-s − 6.27·26-s + 5.65·32-s − 6.66·36-s − 2.52·40-s − 2.49·41-s + 1.20·44-s − 1.49·45-s + 1.17·46-s + 2.33·47-s − 1.42·49-s + 3.95·50-s + 8.87·52-s + 0.269·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 480491 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 480491 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$480491$$    =    $$11^{3} \cdot 19^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{480491} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 1 Selberg data = $(4,\ 480491,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

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

−8.603539619290756001226038948684, −8.271456646397008703308309286986, −7.57431385562905041991919174068, −7.35497151208830631745730081452, −6.38571791919663056094366244020, −6.36261389471308870138602900888, −5.82022083514319120460449276497, −5.31637329804068265476360676633, −4.42771266189087965806263330587, −3.71023016084926681627402828974, −3.06255388161967307495815251846, −2.08501708743688275104095928757, −1.78686819759584429785481099010, −1.00636184286304673122517790617, 0, 1.00636184286304673122517790617, 1.78686819759584429785481099010, 2.08501708743688275104095928757, 3.06255388161967307495815251846, 3.71023016084926681627402828974, 4.42771266189087965806263330587, 5.31637329804068265476360676633, 5.82022083514319120460449276497, 6.36261389471308870138602900888, 6.38571791919663056094366244020, 7.35497151208830631745730081452, 7.57431385562905041991919174068, 8.271456646397008703308309286986, 8.603539619290756001226038948684