# Properties

 Label 4-20e4-1.1-c3e2-0-3 Degree $4$ Conductor $160000$ Sign $1$ Analytic cond. $556.996$ Root an. cond. $4.85806$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 10·9-s − 24·11-s − 200·19-s + 180·29-s − 304·31-s − 876·41-s + 670·49-s + 840·59-s + 1.80e3·61-s − 864·71-s − 320·79-s − 629·81-s − 1.62e3·89-s + 240·99-s − 516·101-s − 1.90e3·109-s − 2.23e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e3·169-s + ⋯
 L(s)  = 1 − 0.370·9-s − 0.657·11-s − 2.41·19-s + 1.15·29-s − 1.76·31-s − 3.33·41-s + 1.95·49-s + 1.85·59-s + 3.78·61-s − 1.44·71-s − 0.455·79-s − 0.862·81-s − 1.92·89-s + 0.243·99-s − 0.508·101-s − 1.66·109-s − 1.67·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.468·169-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$160000$$    =    $$2^{8} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$556.996$$ Root analytic conductor: $$4.85806$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 160000,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.6229674171$$ $$L(\frac12)$$ $$\approx$$ $$0.6229674171$$ $$L(\frac{5}{2})$$ 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 $$1$$
5 $$1$$
good3$C_2^2$ $$1 + 10 T^{2} + p^{6} T^{4}$$
7$C_2^2$ $$1 - 670 T^{2} + p^{6} T^{4}$$
11$C_2$ $$( 1 + 12 T + p^{3} T^{2} )^{2}$$
13$C_2^2$ $$1 - 1030 T^{2} + p^{6} T^{4}$$
17$C_2^2$ $$1 - 5470 T^{2} + p^{6} T^{4}$$
19$C_2$ $$( 1 + 100 T + p^{3} T^{2} )^{2}$$
23$C_2^2$ $$1 - 6910 T^{2} + p^{6} T^{4}$$
29$C_2$ $$( 1 - 90 T + p^{3} T^{2} )^{2}$$
31$C_2$ $$( 1 + 152 T + p^{3} T^{2} )^{2}$$
37$C_2^2$ $$1 - 100150 T^{2} + p^{6} T^{4}$$
41$C_2$ $$( 1 + 438 T + p^{3} T^{2} )^{2}$$
43$C_2^2$ $$1 - 157990 T^{2} + p^{6} T^{4}$$
47$C_2^2$ $$1 - 166030 T^{2} + p^{6} T^{4}$$
53$C_2^2$ $$1 - 248470 T^{2} + p^{6} T^{4}$$
59$C_2$ $$( 1 - 420 T + p^{3} T^{2} )^{2}$$
61$C_2$ $$( 1 - 902 T + p^{3} T^{2} )^{2}$$
67$C_2^2$ $$1 + 447050 T^{2} + p^{6} T^{4}$$
71$C_2$ $$( 1 + 432 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 - 646990 T^{2} + p^{6} T^{4}$$
79$C_2$ $$( 1 + 160 T + p^{3} T^{2} )^{2}$$
83$C_2^2$ $$1 - 1138390 T^{2} + p^{6} T^{4}$$
89$C_2$ $$( 1 + 810 T + p^{3} T^{2} )^{2}$$
97$C_2^2$ $$1 - 602110 T^{2} + p^{6} T^{4}$$
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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

−11.32126578633781093636372224421, −10.40742612228477583123339837037, −10.15791095402599830935992690217, −10.11215882525853187921306998321, −8.931889719329597217922339508713, −8.864491989105973579984721724205, −8.220958256286813827480688066395, −8.193648389541321052519521268420, −7.13531374402991349644883425910, −6.86957754128808352492833102484, −6.50933563664974416561802883032, −5.52432384743826982638099992003, −5.50510706507821248227493189054, −4.76787080715042475526701244230, −3.95786270760744858547981600387, −3.75213779564222301094101297170, −2.65077880984899148337869881597, −2.32184008131005847734259956559, −1.46989980773140512980886815240, −0.25853637319379430745198509969, 0.25853637319379430745198509969, 1.46989980773140512980886815240, 2.32184008131005847734259956559, 2.65077880984899148337869881597, 3.75213779564222301094101297170, 3.95786270760744858547981600387, 4.76787080715042475526701244230, 5.50510706507821248227493189054, 5.52432384743826982638099992003, 6.50933563664974416561802883032, 6.86957754128808352492833102484, 7.13531374402991349644883425910, 8.193648389541321052519521268420, 8.220958256286813827480688066395, 8.864491989105973579984721724205, 8.931889719329597217922339508713, 10.11215882525853187921306998321, 10.15791095402599830935992690217, 10.40742612228477583123339837037, 11.32126578633781093636372224421