# Properties

 Degree 4 Conductor $2^{12} \cdot 5^{2}$ Sign $1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 12·3-s − 50·5-s − 52·7-s + 138·9-s + 560·11-s − 1.38e3·13-s + 600·15-s + 148·17-s − 1.00e3·19-s + 624·21-s + 2.45e3·23-s + 1.87e3·25-s − 4.50e3·27-s − 1.34e3·29-s + 2.24e3·31-s − 6.72e3·33-s + 2.60e3·35-s + 5.94e3·37-s + 1.66e4·39-s + 2.30e4·41-s + 1.76e4·43-s − 6.90e3·45-s + 2.90e3·47-s − 2.69e4·49-s − 1.77e3·51-s + 5.41e3·53-s − 2.80e4·55-s + ⋯
 L(s)  = 1 − 0.769·3-s − 0.894·5-s − 0.401·7-s + 0.567·9-s + 1.39·11-s − 2.27·13-s + 0.688·15-s + 0.124·17-s − 0.635·19-s + 0.308·21-s + 0.966·23-s + 3/5·25-s − 1.18·27-s − 0.295·29-s + 0.420·31-s − 1.07·33-s + 0.358·35-s + 0.713·37-s + 1.75·39-s + 2.14·41-s + 1.45·43-s − 0.507·45-s + 0.192·47-s − 1.60·49-s − 0.0956·51-s + 0.264·53-s − 1.24·55-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 = $$0$$ Selberg data = $$(4,\ 102400,\ (\ :5/2, 5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.314480318$$ $$L(\frac12)$$ $$\approx$$ $$1.314480318$$ $$L(\frac{7}{2})$$ 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$D_{4}$ $$1 + 4 p T + 2 p T^{2} + 4 p^{6} T^{3} + p^{10} T^{4}$$
7$D_{4}$ $$1 + 52 T + 29646 T^{2} + 52 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 - 560 T + 381926 T^{2} - 560 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 1388 T + 1149918 T^{2} + 1388 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 148 T - 795706 T^{2} - 148 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 + 1000 T + 4533462 T^{2} + 1000 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 2452 T + 6569198 T^{2} - 2452 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 1340 T - 8758306 T^{2} + 1340 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 - 2248 T + 57017022 T^{2} - 2248 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 5940 T + 123434318 T^{2} - 5940 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 23076 T + 352280470 T^{2} - 23076 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 - 17684 T + 312898614 T^{2} - 17684 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 - 2908 T + 56660030 T^{2} - 2908 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 5412 T + 693247822 T^{2} - 5412 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 - 62584 T + 2277965606 T^{2} - 62584 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 14108 T + 1110042462 T^{2} + 14108 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 85412 T + 4371910566 T^{2} + 85412 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 + 47208 T + 4011779662 T^{2} + 47208 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 924 p T + 4400780438 T^{2} + 924 p^{6} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 - 65904 T + 3994274078 T^{2} - 65904 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 108724 T + 10572459494 T^{2} - 108724 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 55020 T + 10818978262 T^{2} + 55020 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 147668 T + 11612429670 T^{2} - 147668 p^{5} T^{3} + p^{10} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}