# Properties

 Label 2-20e2-5.4-c5-0-15 Degree $2$ Conductor $400$ Sign $-0.447 - 0.894i$ Analytic cond. $64.1535$ Root an. cond. $8.00958$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·3-s + 192i·7-s + 227·9-s + 148·11-s + 286i·13-s + 1.67e3i·17-s + 1.06e3·19-s − 768·21-s − 2.97e3i·23-s + 1.88e3i·27-s + 3.41e3·29-s + 2.44e3·31-s + 592i·33-s − 182i·37-s − 1.14e3·39-s + ⋯
 L(s)  = 1 + 0.256i·3-s + 1.48i·7-s + 0.934·9-s + 0.368·11-s + 0.469i·13-s + 1.40i·17-s + 0.673·19-s − 0.380·21-s − 1.17i·23-s + 0.496i·27-s + 0.752·29-s + 0.457·31-s + 0.0946i·33-s − 0.0218i·37-s − 0.120·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$400$$    =    $$2^{4} \cdot 5^{2}$$ Sign: $-0.447 - 0.894i$ Analytic conductor: $$64.1535$$ Root analytic conductor: $$8.00958$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{400} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 400,\ (\ :5/2),\ -0.447 - 0.894i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.171368568$$ $$L(\frac12)$$ $$\approx$$ $$2.171368568$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 - 4iT - 243T^{2}$$
7 $$1 - 192iT - 1.68e4T^{2}$$
11 $$1 - 148T + 1.61e5T^{2}$$
13 $$1 - 286iT - 3.71e5T^{2}$$
17 $$1 - 1.67e3iT - 1.41e6T^{2}$$
19 $$1 - 1.06e3T + 2.47e6T^{2}$$
23 $$1 + 2.97e3iT - 6.43e6T^{2}$$
29 $$1 - 3.41e3T + 2.05e7T^{2}$$
31 $$1 - 2.44e3T + 2.86e7T^{2}$$
37 $$1 + 182iT - 6.93e7T^{2}$$
41 $$1 + 9.39e3T + 1.15e8T^{2}$$
43 $$1 - 1.24e3iT - 1.47e8T^{2}$$
47 $$1 + 1.20e4iT - 2.29e8T^{2}$$
53 $$1 - 2.38e4iT - 4.18e8T^{2}$$
59 $$1 + 2.00e4T + 7.14e8T^{2}$$
61 $$1 - 3.23e4T + 8.44e8T^{2}$$
67 $$1 - 6.09e4iT - 1.35e9T^{2}$$
71 $$1 - 3.26e4T + 1.80e9T^{2}$$
73 $$1 + 3.87e4iT - 2.07e9T^{2}$$
79 $$1 + 3.33e4T + 3.07e9T^{2}$$
83 $$1 + 1.67e4iT - 3.93e9T^{2}$$
89 $$1 + 1.01e5T + 5.58e9T^{2}$$
97 $$1 - 1.19e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$