# Properties

 Label 4-160083-1.1-c1e2-0-4 Degree $4$ Conductor $160083$ Sign $-1$ Analytic cond. $10.2070$ Root an. cond. $1.78741$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3-s − 3·4-s − 4·5-s + 4·7-s + 9-s + 3·12-s + 4·15-s + 5·16-s − 4·17-s + 12·20-s − 4·21-s + 2·25-s − 27-s − 12·28-s − 16·35-s − 3·36-s + 12·37-s − 4·41-s − 4·45-s + 16·47-s − 5·48-s + 9·49-s + 4·51-s − 8·59-s − 12·60-s + 4·63-s − 3·64-s + ⋯
 L(s)  = 1 − 0.577·3-s − 3/2·4-s − 1.78·5-s + 1.51·7-s + 1/3·9-s + 0.866·12-s + 1.03·15-s + 5/4·16-s − 0.970·17-s + 2.68·20-s − 0.872·21-s + 2/5·25-s − 0.192·27-s − 2.26·28-s − 2.70·35-s − 1/2·36-s + 1.97·37-s − 0.624·41-s − 0.596·45-s + 2.33·47-s − 0.721·48-s + 9/7·49-s + 0.560·51-s − 1.04·59-s − 1.54·60-s + 0.503·63-s − 3/8·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160083 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160083 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$160083$$    =    $$3^{3} \cdot 7^{2} \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$10.2070$$ Root analytic conductor: $$1.78741$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 160083,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ $$1 + T$$
7$C_2$ $$1 - 4 T + p T^{2}$$
11$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good2$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
5$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 + p T^{2} )^{2}$$
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$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
79$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−8.975261543279013968212899152873, −8.402747192524386456742003548704, −8.092649848612792766954418771551, −7.55113610427119386648970831546, −7.46995816184461358823952634116, −6.55118736577760269645886929040, −5.83278296173294764308357143188, −5.31622482563829752699303340190, −4.65970646396455232760718697930, −4.28499925734034658438358048834, −4.20407903283329927801590362766, −3.43632280535116392331032608080, −2.26752106768048411048763527476, −1.03326922256959689494698211571, 0, 1.03326922256959689494698211571, 2.26752106768048411048763527476, 3.43632280535116392331032608080, 4.20407903283329927801590362766, 4.28499925734034658438358048834, 4.65970646396455232760718697930, 5.31622482563829752699303340190, 5.83278296173294764308357143188, 6.55118736577760269645886929040, 7.46995816184461358823952634116, 7.55113610427119386648970831546, 8.092649848612792766954418771551, 8.402747192524386456742003548704, 8.975261543279013968212899152873