# 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 − 50·5-s − 446·9-s + 292·13-s − 1.40e3·17-s + 1.87e3·25-s + 8.02e3·29-s + 2.95e4·37-s − 8.70e3·41-s + 2.23e4·45-s − 3.16e4·49-s + 3.63e4·53-s + 8.42e4·61-s − 1.46e4·65-s + 5.25e4·73-s + 1.39e5·81-s + 7.02e4·85-s + 6.11e4·89-s + 1.33e5·97-s − 8.55e4·101-s + 2.23e5·109-s + 4.33e5·113-s − 1.30e5·117-s + 1.97e5·121-s − 6.25e4·125-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 0.894·5-s − 1.83·9-s + 0.479·13-s − 1.17·17-s + 3/5·25-s + 1.77·29-s + 3.54·37-s − 0.808·41-s + 1.64·45-s − 1.88·49-s + 1.77·53-s + 2.89·61-s − 0.428·65-s + 1.15·73-s + 2.36·81-s + 1.05·85-s + 0.818·89-s + 1.44·97-s − 0.834·101-s + 1.80·109-s + 3.19·113-s − 0.879·117-s + 1.22·121-s − 0.357·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-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 = $$(4,\ 102400,\ (\ :5/2, 5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$2.188912979$$ $$L(\frac12)$$ $$\approx$$ $$2.188912979$$ $$L(\frac{7}{2})$$ 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$C_2^2$ $$1 + 446 T^{2} + p^{10} T^{4}$$
7$C_2^2$ $$1 + 646 p^{2} T^{2} + p^{10} T^{4}$$
11$C_2^2$ $$1 - 197738 T^{2} + p^{10} T^{4}$$
13$C_2$ $$( 1 - 146 T + p^{5} T^{2} )^{2}$$
17$C_2$ $$( 1 + 702 T + p^{5} T^{2} )^{2}$$
19$C_2^2$ $$1 - 2512762 T^{2} + p^{10} T^{4}$$
23$C_2^2$ $$1 - 3871674 T^{2} + p^{10} T^{4}$$
29$C_2$ $$( 1 - 4010 T + p^{5} T^{2} )^{2}$$
31$C_2^2$ $$1 + 36406942 T^{2} + p^{10} T^{4}$$
37$C_2$ $$( 1 - 14778 T + p^{5} T^{2} )^{2}$$
41$C_2$ $$( 1 + 4350 T + p^{5} T^{2} )^{2}$$
43$C_2^2$ $$1 + 139567886 T^{2} + p^{10} T^{4}$$
47$C_2^2$ $$1 + 422513974 T^{2} + p^{10} T^{4}$$
53$C_2$ $$( 1 - 18154 T + p^{5} T^{2} )^{2}$$
59$C_2^2$ $$1 + 1041470358 T^{2} + p^{10} T^{4}$$
61$C_2$ $$( 1 - 42130 T + p^{5} T^{2} )^{2}$$
67$C_2^2$ $$1 + 2438310974 T^{2} + p^{10} T^{4}$$
71$C_2^2$ $$1 + 1542914862 T^{2} + p^{10} T^{4}$$
73$C_2$ $$( 1 - 26266 T + p^{5} T^{2} )^{2}$$
79$C_2^2$ $$1 + 6078817438 T^{2} + p^{10} T^{4}$$
83$C_2^2$ $$1 - 1875047714 T^{2} + p^{10} T^{4}$$
89$C_2$ $$( 1 - 30570 T + p^{5} T^{2} )^{2}$$
97$C_2$ $$( 1 - 66882 T + p^{5} T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}