# Properties

 Label 4-189728-1.1-c1e2-0-17 Degree $4$ Conductor $189728$ Sign $-1$ Analytic cond. $12.0972$ Root an. cond. $1.86496$ 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 − 8-s − 2·9-s + 16-s + 2·18-s + 6·25-s − 12·29-s − 32-s − 2·36-s − 4·37-s − 7·49-s − 6·50-s − 4·53-s + 12·58-s + 64-s + 2·72-s + 4·74-s − 5·81-s + 7·98-s + 6·100-s + 4·106-s + 20·109-s − 20·113-s − 12·116-s + 121-s + 127-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s + 1/4·16-s + 0.471·18-s + 6/5·25-s − 2.22·29-s − 0.176·32-s − 1/3·36-s − 0.657·37-s − 49-s − 0.848·50-s − 0.549·53-s + 1.57·58-s + 1/8·64-s + 0.235·72-s + 0.464·74-s − 5/9·81-s + 0.707·98-s + 3/5·100-s + 0.388·106-s + 1.91·109-s − 1.88·113-s − 1.11·116-s + 1/11·121-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$189728$$    =    $$2^{5} \cdot 7^{2} \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$12.0972$$ Root analytic conductor: $$1.86496$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 189728,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_1$ $$1 + T$$
7$C_2$ $$1 + p T^{2}$$
11$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good3$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
17$C_2$ $$( 1 - p T^{2} )^{2}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
47$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
53$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
61$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
73$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} )$$
89$C_2$ $$( 1 - p T^{2} )^{2}$$
97$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
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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

−8.858772578072611359526213611131, −8.539034982107098033938033640909, −7.975223574169517042352563461110, −7.46659077481011948645814284815, −7.13989534434007173512612746405, −6.41310543135258797598918521141, −6.13289442032409492126060993659, −5.31164425237226041733774684037, −5.12514243391074363874048515633, −4.19111221198881396430954655287, −3.51782104512240046203049689516, −2.96495953167035953396367353377, −2.19626538646593776391819028602, −1.37922759436015962364017945930, 0, 1.37922759436015962364017945930, 2.19626538646593776391819028602, 2.96495953167035953396367353377, 3.51782104512240046203049689516, 4.19111221198881396430954655287, 5.12514243391074363874048515633, 5.31164425237226041733774684037, 6.13289442032409492126060993659, 6.41310543135258797598918521141, 7.13989534434007173512612746405, 7.46659077481011948645814284815, 7.975223574169517042352563461110, 8.539034982107098033938033640909, 8.858772578072611359526213611131