# Properties

 Degree 4 Conductor $2^{12} \cdot 5^{2}$ Sign $1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 2

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·3-s − 50·5-s − 104·7-s − 158·9-s + 320·11-s + 100·13-s − 400·15-s + 580·17-s + 720·19-s − 832·21-s − 1.68e3·23-s + 1.87e3·25-s − 1.09e3·27-s − 108·29-s − 9.84e3·31-s + 2.56e3·33-s + 5.20e3·35-s − 6.54e3·37-s + 800·39-s − 1.06e4·41-s + 2.56e4·43-s + 7.90e3·45-s − 2.82e4·47-s − 2.52e4·49-s + 4.64e3·51-s − 3.13e4·53-s − 1.60e4·55-s + ⋯
 L(s)  = 1 + 0.513·3-s − 0.894·5-s − 0.802·7-s − 0.650·9-s + 0.797·11-s + 0.164·13-s − 0.459·15-s + 0.486·17-s + 0.457·19-s − 0.411·21-s − 0.665·23-s + 3/5·25-s − 0.289·27-s − 0.0238·29-s − 1.83·31-s + 0.409·33-s + 0.717·35-s − 0.785·37-s + 0.0842·39-s − 0.986·41-s + 2.11·43-s + 0.581·45-s − 1.86·47-s − 1.50·49-s + 0.249·51-s − 1.53·53-s − 0.713·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$102400$$    =    $$2^{12} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{320} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$2$$ Selberg data = $$(4,\ 102400,\ (\ :5/2, 5/2),\ 1)$$ $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
5$C_1$ $$( 1 + p^{2} T )^{2}$$
good3$D_{4}$ $$1 - 8 T + 74 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4}$$
7$D_{4}$ $$1 + 104 T + 36038 T^{2} + 104 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 - 320 T + 319702 T^{2} - 320 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 - 100 T + 297086 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 580 T + 2475814 T^{2} - 580 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 - 720 T + 4633798 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 + 1688 T + 10950502 T^{2} + 1688 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 108 T + 39233214 T^{2} + 108 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 9840 T + 76732702 T^{2} + 9840 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 + 6540 T + 61572814 T^{2} + 6540 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 + 10620 T + 77106582 T^{2} + 10620 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 - 25672 T + 364912302 T^{2} - 25672 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 + 28296 T + 617782998 T^{2} + 28296 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 + 31340 T + 755347886 T^{2} + 31340 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 - 30800 T + 1666896598 T^{2} - 30800 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 24540 T + 1447225822 T^{2} + 24540 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 - 34584 T + 2930656478 T^{2} - 34584 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 12400 T + 1143670702 T^{2} - 12400 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 7180 T + 284279286 T^{2} + 7180 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 + 71840 T + 7430807198 T^{2} + 71840 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 + 31928 T + 5910993662 T^{2} + 31928 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 40748 T + 8570866774 T^{2} + 40748 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 + 190140 T + 19653817414 T^{2} + 190140 p^{5} T^{3} + p^{10} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}