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 + 50·5-s − 306·9-s − 308·13-s + 356·17-s + 1.87e3·25-s − 8.22e3·29-s − 1.48e4·37-s + 1.45e4·41-s − 1.53e4·45-s − 1.43e4·49-s − 6.44e4·53-s − 5.35e4·61-s − 1.54e4·65-s − 3.70e4·73-s + 3.45e4·81-s + 1.78e4·85-s − 2.15e5·89-s − 2.17e5·97-s − 1.18e5·101-s + 2.79e5·109-s − 8.60e4·113-s + 9.42e4·117-s − 2.54e5·121-s + 6.25e4·125-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 + 0.894·5-s − 1.25·9-s − 0.505·13-s + 0.298·17-s + 3/5·25-s − 1.81·29-s − 1.78·37-s + 1.35·41-s − 1.12·45-s − 0.856·49-s − 3.15·53-s − 1.84·61-s − 0.452·65-s − 0.814·73-s + 0.585·81-s + 0.267·85-s − 2.87·89-s − 2.34·97-s − 1.15·101-s + 2.25·109-s − 0.634·113-s + 0.636·117-s − 1.58·121-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 = $$2$$ 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$C_2^2$ $$1 + 34 p^{2} T^{2} + p^{10} T^{4}$$
7$C_2^2$ $$1 + 14394 T^{2} + p^{10} T^{4}$$
11$C_2^2$ $$1 + 254822 T^{2} + p^{10} T^{4}$$
13$C_2$ $$( 1 + 154 T + p^{5} T^{2} )^{2}$$
17$C_2$ $$( 1 - 178 T + p^{5} T^{2} )^{2}$$
19$C_2^2$ $$1 + 4019078 T^{2} + p^{10} T^{4}$$
23$C_2^2$ $$1 + 5934266 T^{2} + p^{10} T^{4}$$
29$C_2$ $$( 1 + 4110 T + p^{5} T^{2} )^{2}$$
31$C_2^2$ $$1 + 47289582 T^{2} + p^{10} T^{4}$$
37$C_2$ $$( 1 + 7442 T + p^{5} T^{2} )^{2}$$
41$C_2$ $$( 1 - 7270 T + p^{5} T^{2} )^{2}$$
43$C_2^2$ $$1 - 26783614 T^{2} + p^{10} T^{4}$$
47$C_2^2$ $$1 + 403777034 T^{2} + p^{10} T^{4}$$
53$C_2$ $$( 1 + 32226 T + p^{5} T^{2} )^{2}$$
59$C_2^2$ $$1 + 270997718 T^{2} + p^{10} T^{4}$$
61$C_2$ $$( 1 + 26770 T + p^{5} T^{2} )^{2}$$
67$C_2^2$ $$1 + 219594834 T^{2} + p^{10} T^{4}$$
71$C_2^2$ $$1 + 681226622 T^{2} + p^{10} T^{4}$$
73$C_2$ $$( 1 + 18534 T + p^{5} T^{2} )^{2}$$
79$C_2^2$ $$1 - 1369983522 T^{2} + p^{10} T^{4}$$
83$C_2^2$ $$1 + 1693436786 T^{2} + p^{10} T^{4}$$
89$C_2$ $$( 1 + 107590 T + p^{5} T^{2} )^{2}$$
97$C_2$ $$( 1 + 108838 T + p^{5} T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}