# Properties

 Label 4-180e2-1.1-c0e2-0-0 Degree $4$ Conductor $32400$ Sign $1$ Analytic cond. $0.00806973$ Root an. cond. $0.299719$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s − 14-s + 15-s − 16-s − 21-s + 23-s − 24-s + 27-s + 29-s − 30-s − 35-s − 40-s + 41-s + 42-s − 2·43-s − 46-s + 47-s + 48-s + 49-s − 54-s + 56-s + ⋯
 L(s)  = 1 − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s − 14-s + 15-s − 16-s − 21-s + 23-s − 24-s + 27-s + 29-s − 30-s − 35-s − 40-s + 41-s + 42-s − 2·43-s − 46-s + 47-s + 48-s + 49-s − 54-s + 56-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$32400$$    =    $$2^{4} \cdot 3^{4} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$0.00806973$$ Root analytic conductor: $$0.299719$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{180} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 32400,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1855957153$$ $$L(\frac12)$$ $$\approx$$ $$0.1855957153$$ $$L(1)$$ 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$C_2$ $$1 + T + T^{2}$$
3$C_2$ $$1 + T + T^{2}$$
5$C_2$ $$1 + T + T^{2}$$
good7$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
11$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
13$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
17$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
23$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
29$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
31$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
37$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
41$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
43$C_2$ $$( 1 + T + T^{2} )^{2}$$
47$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
59$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
61$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
67$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
73$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
79$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
83$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
89$C_2$ $$( 1 + T + T^{2} )^{2}$$
97$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
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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

−13.04264661200903517342027650101, −12.42240255709205703010474074280, −11.81595518153823601415363561867, −11.73600259250111435577981040744, −10.91868286944090228996622822922, −10.90274568229085661893260839840, −10.36180860361938804868043823832, −9.612274076556058093848870083957, −9.144102951616797963774964463606, −8.310140351924579553413081198973, −8.286011076617848398893161331147, −7.72987037568855243508650243723, −6.85589382162551048678061698581, −6.73967792274990793223112557608, −5.44771769571650859231311693028, −5.24204552749825469170490668341, −4.43443877732487047943431891381, −3.96139403068010698317325041094, −2.69175475848050787785423695874, −1.22605061796298853307315365905, 1.22605061796298853307315365905, 2.69175475848050787785423695874, 3.96139403068010698317325041094, 4.43443877732487047943431891381, 5.24204552749825469170490668341, 5.44771769571650859231311693028, 6.73967792274990793223112557608, 6.85589382162551048678061698581, 7.72987037568855243508650243723, 8.286011076617848398893161331147, 8.310140351924579553413081198973, 9.144102951616797963774964463606, 9.612274076556058093848870083957, 10.36180860361938804868043823832, 10.90274568229085661893260839840, 10.91868286944090228996622822922, 11.73600259250111435577981040744, 11.81595518153823601415363561867, 12.42240255709205703010474074280, 13.04264661200903517342027650101