# Properties

 Degree 4 Conductor $5^{2}$ Sign $1$ Motivic weight 7 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 20·2-s + 20·3-s + 120·4-s − 250·5-s + 400·6-s − 100·7-s − 640·8-s + 790·9-s − 5.00e3·10-s + 4.54e3·11-s + 2.40e3·12-s + 3.54e3·13-s − 2.00e3·14-s − 5.00e3·15-s − 1.15e4·16-s − 2.73e4·17-s + 1.58e4·18-s + 3.87e4·19-s − 3.00e4·20-s − 2.00e3·21-s + 9.08e4·22-s − 1.24e5·23-s − 1.28e4·24-s + 4.68e4·25-s + 7.08e4·26-s + 6.73e4·27-s − 1.20e4·28-s + ⋯
 L(s)  = 1 + 1.76·2-s + 0.427·3-s + 0.937·4-s − 0.894·5-s + 0.756·6-s − 0.110·7-s − 0.441·8-s + 0.361·9-s − 1.58·10-s + 1.02·11-s + 0.400·12-s + 0.446·13-s − 0.194·14-s − 0.382·15-s − 0.707·16-s − 1.34·17-s + 0.638·18-s + 1.29·19-s − 0.838·20-s − 0.0471·21-s + 1.81·22-s − 2.12·23-s − 0.189·24-s + 3/5·25-s + 0.789·26-s + 0.658·27-s − 0.103·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$25$$    =    $$5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$7$$ character : induced by $\chi_{5} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 25,\ (\ :7/2, 7/2),\ 1)$ $L(4)$ $\approx$ $2.61347$ $L(\frac12)$ $\approx$ $2.61347$ $L(\frac{9}{2})$ 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$C_1$ $$( 1 + p^{3} T )^{2}$$
good2$D_{4}$ $$1 - 5 p^{2} T + 35 p^{3} T^{2} - 5 p^{9} T^{3} + p^{14} T^{4}$$
3$D_{4}$ $$1 - 20 T - 130 p T^{2} - 20 p^{7} T^{3} + p^{14} T^{4}$$
7$D_{4}$ $$1 + 100 T + 1411250 T^{2} + 100 p^{7} T^{3} + p^{14} T^{4}$$
11$D_{4}$ $$1 - 4544 T + 31976326 T^{2} - 4544 p^{7} T^{3} + p^{14} T^{4}$$
13$D_{4}$ $$1 - 3540 T + 100535470 T^{2} - 3540 p^{7} T^{3} + p^{14} T^{4}$$
17$D_{4}$ $$1 + 27340 T + 901005190 T^{2} + 27340 p^{7} T^{3} + p^{14} T^{4}$$
19$D_{4}$ $$1 - 2040 p T + 113449762 p T^{2} - 2040 p^{8} T^{3} + p^{14} T^{4}$$
23$D_{4}$ $$1 + 124140 T + 10649684530 T^{2} + 124140 p^{7} T^{3} + p^{14} T^{4}$$
29$D_{4}$ $$1 + 72260 T + 6846819118 T^{2} + 72260 p^{7} T^{3} + p^{14} T^{4}$$
31$D_{4}$ $$1 - 306824 T + 77964629966 T^{2} - 306824 p^{7} T^{3} + p^{14} T^{4}$$
37$D_{4}$ $$1 + 123020 T + 144088599870 T^{2} + 123020 p^{7} T^{3} + p^{14} T^{4}$$
41$D_{4}$ $$1 - 264364 T + 161786388886 T^{2} - 264364 p^{7} T^{3} + p^{14} T^{4}$$
43$D_{4}$ $$1 - 423300 T + 446651231050 T^{2} - 423300 p^{7} T^{3} + p^{14} T^{4}$$
47$D_{4}$ $$1 + 105460 T + 858715356610 T^{2} + 105460 p^{7} T^{3} + p^{14} T^{4}$$
53$D_{4}$ $$1 + 2391580 T + 3562552504510 T^{2} + 2391580 p^{7} T^{3} + p^{14} T^{4}$$
59$D_{4}$ $$1 + 1120120 T + 1362334883638 T^{2} + 1120120 p^{7} T^{3} + p^{14} T^{4}$$
61$D_{4}$ $$1 - 2257044 T + 5613447576526 T^{2} - 2257044 p^{7} T^{3} + p^{14} T^{4}$$
67$D_{4}$ $$1 - 4516460 T + 16742087664890 T^{2} - 4516460 p^{7} T^{3} + p^{14} T^{4}$$
71$D_{4}$ $$1 - 621784 T + 17914494152446 T^{2} - 621784 p^{7} T^{3} + p^{14} T^{4}$$
73$D_{4}$ $$1 - 4569060 T + 23424949855030 T^{2} - 4569060 p^{7} T^{3} + p^{14} T^{4}$$
79$D_{4}$ $$1 - 4333040 T + 330830231042 p T^{2} - 4333040 p^{7} T^{3} + p^{14} T^{4}$$
83$D_{4}$ $$1 + 9793020 T + 59971104320890 T^{2} + 9793020 p^{7} T^{3} + p^{14} T^{4}$$
89$D_{4}$ $$1 - 6025620 T + 89865866149558 T^{2} - 6025620 p^{7} T^{3} + p^{14} T^{4}$$
97$D_{4}$ $$1 - 4609540 T + 142930351581510 T^{2} - 4609540 p^{7} T^{3} + p^{14} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}