# Properties

 Label 4-442368-1.1-c1e2-0-0 Degree $4$ Conductor $442368$ Sign $1$ Analytic cond. $28.2057$ Root an. cond. $2.30454$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s + 9-s − 4·11-s + 2·17-s − 8·19-s + 6·25-s − 27-s + 4·33-s − 6·41-s − 12·43-s + 2·49-s − 2·51-s + 8·57-s + 4·59-s + 8·67-s + 4·73-s − 6·75-s + 81-s + 12·83-s + 2·89-s − 16·97-s − 4·99-s + 20·107-s + 10·113-s − 10·121-s + 6·123-s + 127-s + ⋯
 L(s)  = 1 − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.485·17-s − 1.83·19-s + 6/5·25-s − 0.192·27-s + 0.696·33-s − 0.937·41-s − 1.82·43-s + 2/7·49-s − 0.280·51-s + 1.05·57-s + 0.520·59-s + 0.977·67-s + 0.468·73-s − 0.692·75-s + 1/9·81-s + 1.31·83-s + 0.211·89-s − 1.62·97-s − 0.402·99-s + 1.93·107-s + 0.940·113-s − 0.909·121-s + 0.541·123-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$442368$$    =    $$2^{14} \cdot 3^{3}$$ Sign: $1$ Analytic conductor: $$28.2057$$ Root analytic conductor: $$2.30454$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 442368,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9790939420$$ $$L(\frac12)$$ $$\approx$$ $$0.9790939420$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_1$ $$1 + T$$
good5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
13$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
23$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 12 T + p T^{2} )$$
47$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
71$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2^2$ $$1 + 46 T^{2} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
89$C_2$$\times$$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
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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

−8.504260956025760287998725072767, −8.174291916014102954935419558366, −7.76139333301962742643024545645, −7.08634357061706526731867665475, −6.66435669311267245708484217317, −6.43546144631690904479700213719, −5.64636536881997697230542212926, −5.34318718203313404680289768503, −4.79196669428260128340004223891, −4.43417520195675120973707277560, −3.64043174116040341978914976440, −3.11035369176159785408075083066, −2.35010546621296096213549478531, −1.74602073288839591527261259305, −0.54245484583042170863001187940, 0.54245484583042170863001187940, 1.74602073288839591527261259305, 2.35010546621296096213549478531, 3.11035369176159785408075083066, 3.64043174116040341978914976440, 4.43417520195675120973707277560, 4.79196669428260128340004223891, 5.34318718203313404680289768503, 5.64636536881997697230542212926, 6.43546144631690904479700213719, 6.66435669311267245708484217317, 7.08634357061706526731867665475, 7.76139333301962742643024545645, 8.174291916014102954935419558366, 8.504260956025760287998725072767