# Properties

 Degree 4 Conductor $2^{4} \cdot 5^{2}$ Sign $1$ Motivic weight 9 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 260·3-s − 1.25e3·5-s − 380·7-s + 3.15e4·9-s + 1.02e5·11-s + 1.79e5·13-s + 3.25e5·15-s + 3.16e5·17-s + 1.37e5·19-s + 9.88e4·21-s − 6.65e5·23-s + 1.17e6·25-s − 3.95e6·27-s − 6.89e6·29-s + 2.91e5·31-s − 2.67e7·33-s + 4.75e5·35-s + 1.12e7·37-s − 4.65e7·39-s + 2.97e7·41-s − 1.17e7·43-s − 3.94e7·45-s + 6.24e7·47-s + 1.56e7·49-s − 8.21e7·51-s + 9.41e6·53-s − 1.28e8·55-s + ⋯
 L(s)  = 1 − 1.85·3-s − 0.894·5-s − 0.0598·7-s + 1.60·9-s + 2.11·11-s + 1.73·13-s + 1.65·15-s + 0.917·17-s + 0.241·19-s + 0.110·21-s − 0.495·23-s + 3/5·25-s − 1.43·27-s − 1.80·29-s + 0.0567·31-s − 3.92·33-s + 0.0535·35-s + 0.987·37-s − 3.22·39-s + 1.64·41-s − 0.522·43-s − 1.43·45-s + 1.86·47-s + 0.388·49-s − 1.70·51-s + 0.163·53-s − 1.89·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$400$$    =    $$2^{4} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$9$$ character : induced by $\chi_{20} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 400,\ (\ :9/2, 9/2),\ 1)$$ $$L(5)$$ $$\approx$$ $$1.11465$$ $$L(\frac12)$$ $$\approx$$ $$1.11465$$ $$L(\frac{11}{2})$$ 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$C_1$ $$( 1 + p^{4} T )^{2}$$
good3$D_{4}$ $$1 + 260 T + 12014 p T^{2} + 260 p^{9} T^{3} + p^{18} T^{4}$$
7$D_{4}$ $$1 + 380 T - 2220450 p T^{2} + 380 p^{9} T^{3} + p^{18} T^{4}$$
11$D_{4}$ $$1 - 102720 T + 7335543382 T^{2} - 102720 p^{9} T^{3} + p^{18} T^{4}$$
13$D_{4}$ $$1 - 1060 p^{2} T + 22798610142 T^{2} - 1060 p^{11} T^{3} + p^{18} T^{4}$$
17$D_{4}$ $$1 - 316020 T + 259693705798 T^{2} - 316020 p^{9} T^{3} + p^{18} T^{4}$$
19$D_{4}$ $$1 - 137272 T + 111610161654 T^{2} - 137272 p^{9} T^{3} + p^{18} T^{4}$$
23$D_{4}$ $$1 + 665460 T + 2886450615250 T^{2} + 665460 p^{9} T^{3} + p^{18} T^{4}$$
29$D_{4}$ $$1 + 6893748 T + 40195999658014 T^{2} + 6893748 p^{9} T^{3} + p^{18} T^{4}$$
31$D_{4}$ $$1 - 291832 T + 38964935800398 T^{2} - 291832 p^{9} T^{3} + p^{18} T^{4}$$
37$D_{4}$ $$1 - 11261380 T + 218879982937230 T^{2} - 11261380 p^{9} T^{3} + p^{18} T^{4}$$
41$D_{4}$ $$1 - 29773452 T + 771012402449398 T^{2} - 29773452 p^{9} T^{3} + p^{18} T^{4}$$
43$D_{4}$ $$1 + 11708180 T + 838769843899386 T^{2} + 11708180 p^{9} T^{3} + p^{18} T^{4}$$
47$D_{4}$ $$1 - 62493300 T + 3177958884734338 T^{2} - 62493300 p^{9} T^{3} + p^{18} T^{4}$$
53$D_{4}$ $$1 - 9417780 T + 5708185761526990 T^{2} - 9417780 p^{9} T^{3} + p^{18} T^{4}$$
59$D_{4}$ $$1 + 92930856 T + 16656477955483462 T^{2} + 92930856 p^{9} T^{3} + p^{18} T^{4}$$
61$D_{4}$ $$1 - 195673924 T + 22434263296171326 T^{2} - 195673924 p^{9} T^{3} + p^{18} T^{4}$$
67$D_{4}$ $$1 + 219767420 T + 65652945987990090 T^{2} + 219767420 p^{9} T^{3} + p^{18} T^{4}$$
71$D_{4}$ $$1 - 311207016 T + 76405636625293726 T^{2} - 311207016 p^{9} T^{3} + p^{18} T^{4}$$
73$D_{4}$ $$1 + 99224060 T + 35402447061205782 T^{2} + 99224060 p^{9} T^{3} + p^{18} T^{4}$$
79$D_{4}$ $$1 - 542261776 T + 313115996157615582 T^{2} - 542261776 p^{9} T^{3} + p^{18} T^{4}$$
83$D_{4}$ $$1 + 1256915700 T + 768086791626261130 T^{2} + 1256915700 p^{9} T^{3} + p^{18} T^{4}$$
89$D_{4}$ $$1 + 462291852 T + 159603168035249494 T^{2} + 462291852 p^{9} T^{3} + p^{18} T^{4}$$
97$D_{4}$ $$1 - 1671716740 T + 2048690578856969670 T^{2} - 1671716740 p^{9} T^{3} + p^{18} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}