# Properties

 Degree 4 Conductor $2^{6} \cdot 3^{4} \cdot 5^{4}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s − 2·3-s − 2·6-s − 8-s + 3·9-s − 2·11-s − 16-s − 4·17-s + 3·18-s − 2·19-s − 2·22-s + 2·24-s − 4·27-s + 4·33-s − 4·34-s − 2·38-s + 41-s − 43-s + 2·48-s − 49-s + 8·51-s − 4·54-s + 4·57-s + 59-s + 64-s + 4·66-s − 67-s + ⋯
 L(s)  = 1 + 2-s − 2·3-s − 2·6-s − 8-s + 3·9-s − 2·11-s − 16-s − 4·17-s + 3·18-s − 2·19-s − 2·22-s + 2·24-s − 4·27-s + 4·33-s − 4·34-s − 2·38-s + 41-s − 43-s + 2·48-s − 49-s + 8·51-s − 4·54-s + 4·57-s + 59-s + 64-s + 4·66-s − 67-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$3240000$$    =    $$2^{6} \cdot 3^{4} \cdot 5^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{1800} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 3240000,\ (\ :0, 0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.07231841195$ $L(\frac12)$ $\approx$ $0.07231841195$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5\}$, $$F_p$$ is a polynomial of degree 4. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 - T + T^{2}$$
3$C_1$ $$( 1 + T )^{2}$$
5 $$1$$
good7$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
11$C_2$ $$( 1 + T + T^{2} )^{2}$$
13$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
17$C_1$ $$( 1 + T )^{4}$$
19$C_2$ $$( 1 + T + T^{2} )^{2}$$
23$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
29$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
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_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
47$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
59$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
61$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
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$C_2$ $$( 1 - T + T^{2} )^{2}$$
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_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
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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

−9.910023944342792120838150622733, −9.221672433493689494542748373485, −8.931788012563920906567232087794, −8.484988356238490620612424190935, −8.092650685060006443180412257156, −7.52731592307891002553681013779, −6.91336963670413770661341706546, −6.62827361643327361832839757443, −6.42028049841365022151502090050, −6.10830087856580903077164341736, −5.36970526932147481326199900939, −5.31283448836759103443258226680, −4.60099810867796520080950289180, −4.58211161765208124931401182762, −4.18826557465494761172551114441, −3.72573923948335484107591889285, −2.53060944909435527588667559895, −2.44656023537320876925877225840, −1.77606184994408508858255127027, −0.18068456797024635616045494218, 0.18068456797024635616045494218, 1.77606184994408508858255127027, 2.44656023537320876925877225840, 2.53060944909435527588667559895, 3.72573923948335484107591889285, 4.18826557465494761172551114441, 4.58211161765208124931401182762, 4.60099810867796520080950289180, 5.31283448836759103443258226680, 5.36970526932147481326199900939, 6.10830087856580903077164341736, 6.42028049841365022151502090050, 6.62827361643327361832839757443, 6.91336963670413770661341706546, 7.52731592307891002553681013779, 8.092650685060006443180412257156, 8.484988356238490620612424190935, 8.931788012563920906567232087794, 9.221672433493689494542748373485, 9.910023944342792120838150622733