# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 20·4-s − 90·5-s + 90·9-s + 504·11-s − 624·16-s − 440·19-s − 1.80e3·20-s + 4.97e3·25-s − 1.38e4·29-s + 1.35e4·31-s + 1.80e3·36-s − 396·41-s + 1.00e4·44-s − 8.10e3·45-s + 3.00e4·49-s − 4.53e4·55-s − 4.93e4·59-s − 1.13e4·61-s − 3.29e4·64-s + 1.06e5·71-s − 8.80e3·76-s + 1.03e5·79-s + 5.61e4·80-s − 5.09e4·81-s − 1.99e4·89-s + 3.96e4·95-s + 4.53e4·99-s + ⋯
 L(s)  = 1 + 5/8·4-s − 1.60·5-s + 0.370·9-s + 1.25·11-s − 0.609·16-s − 0.279·19-s − 1.00·20-s + 1.59·25-s − 3.06·29-s + 2.52·31-s + 0.231·36-s − 0.0367·41-s + 0.784·44-s − 0.596·45-s + 1.78·49-s − 2.02·55-s − 1.84·59-s − 0.392·61-s − 1.00·64-s + 2.51·71-s − 0.174·76-s + 1.87·79-s + 0.981·80-s − 0.862·81-s − 0.267·89-s + 0.450·95-s + 0.465·99-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$25$$    =    $$5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{5} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 25,\ (\ :5/2, 5/2),\ 1)$ $L(3)$ $\approx$ $0.854944$ $L(\frac12)$ $\approx$ $0.854944$ $L(\frac{7}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 4. If $p = 5$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ $$1 + 18 p T + p^{5} T^{2}$$
good2$V_4$ $$1 - 5 p^{2} T^{2} + p^{10} T^{4}$$
3$C_2$ $$( 1 - 8 p T + p^{5} T^{2} )( 1 + 8 p T + p^{5} T^{2} )$$
7$V_4$ $$1 - 30050 T^{2} + p^{10} T^{4}$$
11$C_2$ $$( 1 - 252 T + p^{5} T^{2} )^{2}$$
13$V_4$ $$1 - 728330 T^{2} + p^{10} T^{4}$$
17$V_4$ $$1 - 2363810 T^{2} + p^{10} T^{4}$$
19$C_2$ $$( 1 + 220 T + p^{5} T^{2} )^{2}$$
23$V_4$ $$1 - 6946370 T^{2} + p^{10} T^{4}$$
29$C_2$ $$( 1 + 6930 T + p^{5} T^{2} )^{2}$$
31$C_2$ $$( 1 - 6752 T + p^{5} T^{2} )^{2}$$
37$V_4$ $$1 + 56462470 T^{2} + p^{10} T^{4}$$
41$C_2$ $$( 1 + 198 T + p^{5} T^{2} )^{2}$$
43$V_4$ $$1 - 293842250 T^{2} + p^{10} T^{4}$$
47$V_4$ $$1 - 347593490 T^{2} + p^{10} T^{4}$$
53$V_4$ $$1 - 802472090 T^{2} + p^{10} T^{4}$$
59$C_2$ $$( 1 + 24660 T + p^{5} T^{2} )^{2}$$
61$C_2$ $$( 1 + 5698 T + p^{5} T^{2} )^{2}$$
67$V_4$ $$1 - 795787610 T^{2} + p^{10} T^{4}$$
71$C_2$ $$( 1 - 53352 T + p^{5} T^{2} )^{2}$$
73$V_4$ $$1 + 883886830 T^{2} + p^{10} T^{4}$$
79$C_2$ $$( 1 - 51920 T + p^{5} T^{2} )^{2}$$
83$V_4$ $$1 - 4053674810 T^{2} + p^{10} T^{4}$$
89$C_2$ $$( 1 + 9990 T + p^{5} T^{2} )^{2}$$
97$V_4$ $$1 - 6923133890 T^{2} + p^{10} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}