# Properties

 Degree 4 Conductor $2^{10} \cdot 5^{4}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 2·7-s + 2·9-s − 4·21-s − 2·23-s − 2·27-s + 2·43-s − 2·47-s + 2·49-s + 4·63-s + 2·67-s + 4·69-s + 3·81-s + 2·83-s + 4·101-s + 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s − 4·147-s + 149-s + 151-s + ⋯
 L(s)  = 1 − 2·3-s + 2·7-s + 2·9-s − 4·21-s − 2·23-s − 2·27-s + 2·43-s − 2·47-s + 2·49-s + 4·63-s + 2·67-s + 4·69-s + 3·81-s + 2·83-s + 4·101-s + 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s − 4·147-s + 149-s + 151-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$640000$$    =    $$2^{10} \cdot 5^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{800} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 640000,\ (\ :0, 0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.5353003331$ $L(\frac12)$ $\approx$ $0.5353003331$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p$$ is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 $$1$$
5 $$1$$
good3$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
7$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
11$C_2$ $$( 1 + T^{2} )^{2}$$
13$C_2^2$ $$1 + T^{4}$$
17$C_2^2$ $$1 + T^{4}$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
23$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
29$C_2$ $$( 1 + T^{2} )^{2}$$
31$C_2$ $$( 1 + T^{2} )^{2}$$
37$C_2^2$ $$1 + T^{4}$$
41$C_2$ $$( 1 + T^{2} )^{2}$$
43$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
47$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
53$C_2^2$ $$1 + T^{4}$$
59$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
61$C_2$ $$( 1 + T^{2} )^{2}$$
67$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
71$C_2$ $$( 1 + T^{2} )^{2}$$
73$C_2^2$ $$1 + T^{4}$$
79$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
83$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
89$C_2$ $$( 1 + T^{2} )^{2}$$
97$C_2^2$ $$1 + T^{4}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−10.75221685985421843874095155583, −10.52182806991915867942435966588, −10.03268865522979731250990617072, −9.516285885690007470604571955858, −9.099879670111464566614379913831, −8.354335030273642507666476163441, −7.987549958995614787489586635325, −7.79309233068281090636853498539, −7.23989387664129141346164886590, −6.64426959537030984101543600395, −6.05031302786525693150766574450, −5.94671590199537883366208087967, −5.35118785830813710866323432898, −4.90110252422022193942073224709, −4.68241429286250276903957052486, −4.04655376704796702448425754963, −3.53643054532470293052896772089, −2.16891100401709272297954391108, −1.91087069452013848845759771743, −0.912779280286979310418994085698, 0.912779280286979310418994085698, 1.91087069452013848845759771743, 2.16891100401709272297954391108, 3.53643054532470293052896772089, 4.04655376704796702448425754963, 4.68241429286250276903957052486, 4.90110252422022193942073224709, 5.35118785830813710866323432898, 5.94671590199537883366208087967, 6.05031302786525693150766574450, 6.64426959537030984101543600395, 7.23989387664129141346164886590, 7.79309233068281090636853498539, 7.987549958995614787489586635325, 8.354335030273642507666476163441, 9.099879670111464566614379913831, 9.516285885690007470604571955858, 10.03268865522979731250990617072, 10.52182806991915867942435966588, 10.75221685985421843874095155583