# Properties

 Degree $4$ Conductor $2500$ Sign $1$ Motivic weight $5$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·4-s + 450·9-s + 384·11-s + 256·16-s + 5.48e3·19-s − 1.18e4·29-s − 1.37e4·31-s − 7.20e3·36-s − 756·41-s − 6.14e3·44-s + 1.96e4·49-s + 6.99e4·59-s − 1.96e4·61-s − 4.09e3·64-s + 1.40e5·71-s − 8.76e4·76-s − 9.04e3·79-s + 1.43e5·81-s − 7.69e4·89-s + 1.72e5·99-s + 1.55e5·101-s − 4.13e5·109-s + 1.89e5·116-s − 2.11e5·121-s + 2.19e5·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 − 1/2·4-s + 1.85·9-s + 0.956·11-s + 1/4·16-s + 3.48·19-s − 2.60·29-s − 2.56·31-s − 0.925·36-s − 0.0702·41-s − 0.478·44-s + 1.17·49-s + 2.61·59-s − 0.677·61-s − 1/8·64-s + 3.30·71-s − 1.74·76-s − 0.162·79-s + 2.42·81-s − 1.03·89-s + 1.77·99-s + 1.51·101-s − 3.33·109-s + 1.30·116-s − 1.31·121-s + 1.28·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2500$$    =    $$2^{2} \cdot 5^{4}$$ Sign: $1$ Motivic weight: $$5$$ Character: induced by $\chi_{50} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 2500,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.55589$$ $$L(\frac12)$$ $$\approx$$ $$2.55589$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + p^{4} T^{2}$$
5 $$1$$
good3$C_2^2$ $$1 - 50 p^{2} T^{2} + p^{10} T^{4}$$
7$C_2^2$ $$1 - 19690 T^{2} + p^{10} T^{4}$$
11$C_2$ $$( 1 - 192 T + p^{5} T^{2} )^{2}$$
13$C_2^2$ $$1 + 480650 T^{2} + p^{10} T^{4}$$
17$C_2^2$ $$1 - 2259070 T^{2} + p^{10} T^{4}$$
19$C_2$ $$( 1 - 2740 T + p^{5} T^{2} )^{2}$$
23$C_2^2$ $$1 - 10420330 T^{2} + p^{10} T^{4}$$
29$C_2$ $$( 1 + 5910 T + p^{5} T^{2} )^{2}$$
31$C_2$ $$( 1 + 6868 T + p^{5} T^{2} )^{2}$$
37$C_2^2$ $$1 - 108239590 T^{2} + p^{10} T^{4}$$
41$C_2$ $$( 1 + 378 T + p^{5} T^{2} )^{2}$$
43$C_2^2$ $$1 - 288092530 T^{2} + p^{10} T^{4}$$
47$C_2^2$ $$1 - 286503130 T^{2} + p^{10} T^{4}$$
53$C_2^2$ $$1 - 752228710 T^{2} + p^{10} T^{4}$$
59$C_2$ $$( 1 - 34980 T + p^{5} T^{2} )^{2}$$
61$C_2$ $$( 1 + 9838 T + p^{5} T^{2} )^{2}$$
67$C_2^2$ $$1 - 1563076930 T^{2} + p^{10} T^{4}$$
71$C_2$ $$( 1 - 70212 T + p^{5} T^{2} )^{2}$$
73$C_2^2$ $$1 - 3662758990 T^{2} + p^{10} T^{4}$$
79$C_2$ $$( 1 + 4520 T + p^{5} T^{2} )^{2}$$
83$C_2^2$ $$1 + 4019056190 T^{2} + p^{10} T^{4}$$
89$C_2$ $$( 1 + 38490 T + p^{5} T^{2} )^{2}$$
97$C_2^2$ $$1 - 17171001790 T^{2} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$