# Properties

 Label 2-50e2-100.59-c0-0-1 Degree $2$ Conductor $2500$ Sign $-0.992 - 0.125i$ Analytic cond. $1.24766$ Root an. cond. $1.11698$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.809 + 0.587i)2-s + (0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s − 0.618·7-s + (−0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s + (−1.30 + 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s − 1.61·18-s + (−0.309 − 0.951i)21-s + (0.5 + 0.363i)23-s − 1.61·24-s + (−0.809 − 0.587i)27-s + (−0.190 − 0.587i)28-s + ⋯
 L(s)  = 1 + (0.809 + 0.587i)2-s + (0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s − 0.618·7-s + (−0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s + (−1.30 + 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s − 1.61·18-s + (−0.309 − 0.951i)21-s + (0.5 + 0.363i)23-s − 1.61·24-s + (−0.809 − 0.587i)27-s + (−0.190 − 0.587i)28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2500$$    =    $$2^{2} \cdot 5^{4}$$ Sign: $-0.992 - 0.125i$ Analytic conductor: $$1.24766$$ Root analytic conductor: $$1.11698$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2500} (1499, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2500,\ (\ :0),\ -0.992 - 0.125i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.977445347$$ $$L(\frac12)$$ $$\approx$$ $$1.977445347$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.809 - 0.587i)T$$
5 $$1$$
good3 $$1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}$$
7 $$1 + 0.618T + T^{2}$$
11 $$1 + (-0.309 - 0.951i)T^{2}$$
13 $$1 + (-0.309 + 0.951i)T^{2}$$
17 $$1 + (0.809 + 0.587i)T^{2}$$
19 $$1 + (0.809 + 0.587i)T^{2}$$
23 $$1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}$$
29 $$1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}$$
31 $$1 + (0.809 + 0.587i)T^{2}$$
37 $$1 + (-0.309 + 0.951i)T^{2}$$
41 $$1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}$$
43 $$1 - 1.61T + T^{2}$$
47 $$1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}$$
53 $$1 + (0.809 - 0.587i)T^{2}$$
59 $$1 + (-0.309 + 0.951i)T^{2}$$
61 $$1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}$$
67 $$1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2}$$
71 $$1 + (0.809 - 0.587i)T^{2}$$
73 $$1 + (-0.309 - 0.951i)T^{2}$$
79 $$1 + (0.809 - 0.587i)T^{2}$$
83 $$1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2}$$
89 $$1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}$$
97 $$1 + (0.809 - 0.587i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$