# Properties

 Label 4-728e2-1.1-c1e2-0-17 Degree $4$ Conductor $529984$ Sign $-1$ Analytic cond. $33.7922$ Root an. cond. $2.41103$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s − 4-s − 4·5-s − 3·8-s − 9-s − 4·10-s + 4·11-s − 16-s − 18-s + 4·20-s + 4·22-s + 2·25-s + 5·32-s + 36-s + 6·37-s + 12·40-s − 4·44-s + 4·45-s + 49-s + 2·50-s − 16·55-s + 8·59-s + 7·64-s − 12·67-s + 3·72-s + 6·74-s + 4·80-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 1/3·9-s − 1.26·10-s + 1.20·11-s − 1/4·16-s − 0.235·18-s + 0.894·20-s + 0.852·22-s + 2/5·25-s + 0.883·32-s + 1/6·36-s + 0.986·37-s + 1.89·40-s − 0.603·44-s + 0.596·45-s + 1/7·49-s + 0.282·50-s − 2.15·55-s + 1.04·59-s + 7/8·64-s − 1.46·67-s + 0.353·72-s + 0.697·74-s + 0.447·80-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$529984$$    =    $$2^{6} \cdot 7^{2} \cdot 13^{2}$$ Sign: $-1$ Analytic conductor: $$33.7922$$ Root analytic conductor: $$2.41103$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 529984,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad2$C_2$ $$1 - T + p T^{2}$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
13$C_2$ $$1 + p T^{2}$$
good3$C_2^2$ $$1 + T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
11$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )$$
17$C_2$ $$( 1 + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
29$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 35 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
41$C_2^2$ $$1 + 63 T^{2} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
47$C_2^2$ $$1 + 21 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 74 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
61$C_2^2$ $$1 - 13 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
71$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
73$C_2^2$ $$1 + 7 T^{2} + p^{2} T^{4}$$
79$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
83$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
89$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
97$C_2^2$ $$1 - 41 T^{2} + p^{2} T^{4}$$
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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

−8.212614976415868426881594102776, −7.894014480689558453525107226025, −7.29852027322316878912798387936, −6.99563515973229159888948220181, −6.30562261272351606630460835990, −5.87940791339678818682803724579, −5.49112702887785723394014702678, −4.58744773421140665205821467277, −4.41261204778431246038142296531, −4.00293070751956413301583998965, −3.43279500498160272058752213634, −3.15402500784017426894696132013, −2.18685260072097675439858465857, −0.992849369106282281087130020413, 0, 0.992849369106282281087130020413, 2.18685260072097675439858465857, 3.15402500784017426894696132013, 3.43279500498160272058752213634, 4.00293070751956413301583998965, 4.41261204778431246038142296531, 4.58744773421140665205821467277, 5.49112702887785723394014702678, 5.87940791339678818682803724579, 6.30562261272351606630460835990, 6.99563515973229159888948220181, 7.29852027322316878912798387936, 7.894014480689558453525107226025, 8.212614976415868426881594102776