# Properties

 Label 4-3920e2-1.1-c0e2-0-6 Degree $4$ Conductor $15366400$ Sign $1$ Analytic cond. $3.82724$ Root an. cond. $1.39869$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 5-s + 9-s + 4·29-s + 4·41-s − 45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s − 121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 5-s + 9-s + 4·29-s + 4·41-s − 45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s − 121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$15366400$$    =    $$2^{8} \cdot 5^{2} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$3.82724$$ Root analytic conductor: $$1.39869$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3920} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 15366400,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.545164227$$ $$L(\frac12)$$ $$\approx$$ $$1.545164227$$ $$L(1)$$ 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 $$1$$
5$C_2$ $$1 + T + T^{2}$$
7 $$1$$
good3$C_2^2$ $$1 - T^{2} + T^{4}$$
11$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
13$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
17$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
19$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
23$C_2^2$ $$1 - T^{2} + T^{4}$$
29$C_1$ $$( 1 - T )^{4}$$
31$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
37$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
41$C_1$ $$( 1 - T )^{4}$$
43$C_2$ $$( 1 + T^{2} )^{2}$$
47$C_2^2$ $$1 - T^{2} + T^{4}$$
53$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
59$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
61$C_2$ $$( 1 + T + T^{2} )^{2}$$
67$C_2^2$ $$1 - T^{2} + T^{4}$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
73$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
79$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
83$C_2$ $$( 1 + T^{2} )^{2}$$
89$C_2$ $$( 1 - T + T^{2} )^{2}$$
97$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
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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

−8.702492116265349537293042649560, −8.476584581521341309333890046944, −7.955275465023248645659869494878, −7.77915737582202487182340114251, −7.32983875188193982980792942525, −7.24802992507568953803685822014, −6.44553819269419844744212034115, −6.42174566478572999440638066040, −6.00618644976148158386440807334, −5.51556473355296007245095231125, −4.71309788299406934760416163815, −4.68396227160948575842711348784, −4.31122266114823230233121323007, −4.06327471098678505270898339461, −3.21928571733310263959036936247, −3.17364617860842158216506235416, −2.45327454368982615807571498187, −2.08894897484033772793513713653, −0.990578571973092936812892467527, −0.976050005442044870313045094085, 0.976050005442044870313045094085, 0.990578571973092936812892467527, 2.08894897484033772793513713653, 2.45327454368982615807571498187, 3.17364617860842158216506235416, 3.21928571733310263959036936247, 4.06327471098678505270898339461, 4.31122266114823230233121323007, 4.68396227160948575842711348784, 4.71309788299406934760416163815, 5.51556473355296007245095231125, 6.00618644976148158386440807334, 6.42174566478572999440638066040, 6.44553819269419844744212034115, 7.24802992507568953803685822014, 7.32983875188193982980792942525, 7.77915737582202487182340114251, 7.955275465023248645659869494878, 8.476584581521341309333890046944, 8.702492116265349537293042649560