# Properties

 Label 4-84e2-1.1-c5e2-0-3 Degree $4$ Conductor $7056$ Sign $1$ Analytic cond. $181.501$ Root an. cond. $3.67045$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 18·3-s − 6·5-s − 98·7-s + 243·9-s − 90·11-s + 768·13-s + 108·15-s + 1.92e3·17-s + 2.24e3·19-s + 1.76e3·21-s + 6.35e3·23-s − 654·25-s − 2.91e3·27-s + 1.05e4·29-s − 3.31e3·31-s + 1.62e3·33-s + 588·35-s + 2.10e3·37-s − 1.38e4·39-s + 1.26e3·41-s − 5.76e3·43-s − 1.45e3·45-s + 1.56e4·47-s + 7.20e3·49-s − 3.46e4·51-s + 1.65e4·53-s + 540·55-s + ⋯
 L(s)  = 1 − 1.15·3-s − 0.107·5-s − 0.755·7-s + 9-s − 0.224·11-s + 1.26·13-s + 0.123·15-s + 1.61·17-s + 1.42·19-s + 0.872·21-s + 2.50·23-s − 0.209·25-s − 0.769·27-s + 2.33·29-s − 0.618·31-s + 0.258·33-s + 0.0811·35-s + 0.252·37-s − 1.45·39-s + 0.117·41-s − 0.475·43-s − 0.107·45-s + 1.03·47-s + 3/7·49-s − 1.86·51-s + 0.807·53-s + 0.0240·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$7056$$    =    $$2^{4} \cdot 3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$181.501$$ Root analytic conductor: $$3.67045$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 7056,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.956728909$$ $$L(\frac12)$$ $$\approx$$ $$1.956728909$$ $$L(\frac{7}{2})$$ 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)$
bad2 $$1$$
3$C_1$ $$( 1 + p^{2} T )^{2}$$
7$C_1$ $$( 1 + p^{2} T )^{2}$$
good5$D_{4}$ $$1 + 6 T + 138 p T^{2} + 6 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 + 90 T + 51246 T^{2} + 90 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 - 768 T + 689558 T^{2} - 768 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 1926 T + 3761514 T^{2} - 1926 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 - 2248 T + 3007830 T^{2} - 2248 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 6354 T + 22960446 T^{2} - 6354 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 - 10572 T + 60053694 T^{2} - 10572 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 3312 T + 31130942 T^{2} + 3312 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 2104 T + 14492118 T^{2} - 2104 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 1266 T + 226048450 T^{2} - 1266 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 5768 T - 18440058 T^{2} + 5768 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 - 15612 T + 373470814 T^{2} - 15612 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 16512 T + 599616358 T^{2} - 16512 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 13140 T + 1459090998 T^{2} + 13140 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 5796 T + 1309453982 T^{2} + 5796 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 56116 T + 2298831942 T^{2} + 56116 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 11022 T - 822078402 T^{2} - 11022 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 85384 T + 5800143006 T^{2} + 85384 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 + 19620 T + 6216467102 T^{2} + 19620 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 + 44424 T + 7801188630 T^{2} + 44424 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 - 211218 T + 21434789410 T^{2} - 211218 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 44864 T + 3170652414 T^{2} - 44864 p^{5} T^{3} + p^{10} T^{4}$$
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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

−13.37449393757759025822871781524, −13.04589591097800448108816067438, −12.36927526074276033777729526304, −11.95372122001901036564881757161, −11.45809232237543543265642907831, −10.88637087035719379233582877228, −10.14131161289535531956222982294, −10.11197894600984929543742669939, −9.020374215300732690055887067713, −8.728896099444783214461133209925, −7.44464416532910332841273569285, −7.38488820157283642554665432599, −6.37066729040917812049820557988, −5.94729515074259491783820941558, −5.24757818509928590157425842175, −4.64553794718278098278065448766, −3.44196985942801103584946040914, −3.02176397617772128275971656079, −1.14117276929874442954233315634, −0.812487711873018734811142710361, 0.812487711873018734811142710361, 1.14117276929874442954233315634, 3.02176397617772128275971656079, 3.44196985942801103584946040914, 4.64553794718278098278065448766, 5.24757818509928590157425842175, 5.94729515074259491783820941558, 6.37066729040917812049820557988, 7.38488820157283642554665432599, 7.44464416532910332841273569285, 8.728896099444783214461133209925, 9.020374215300732690055887067713, 10.11197894600984929543742669939, 10.14131161289535531956222982294, 10.88637087035719379233582877228, 11.45809232237543543265642907831, 11.95372122001901036564881757161, 12.36927526074276033777729526304, 13.04589591097800448108816067438, 13.37449393757759025822871781524