# Properties

 Label 2-1100-5.4-c5-0-19 Degree $2$ Conductor $1100$ Sign $-0.447 - 0.894i$ Analytic cond. $176.422$ Root an. cond. $13.2824$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 7i·3-s + 50i·7-s + 194·9-s + 121·11-s − 380i·13-s + 1.15e3i·17-s + 1.82e3·19-s − 350·21-s + 3.59e3i·23-s + 3.05e3i·27-s − 8.03e3·29-s − 2.94e3·31-s + 847i·33-s − 6.97e3i·37-s + 2.66e3·39-s + ⋯
 L(s)  = 1 + 0.449i·3-s + 0.385i·7-s + 0.798·9-s + 0.301·11-s − 0.623i·13-s + 0.968i·17-s + 1.15·19-s − 0.173·21-s + 1.41i·23-s + 0.807i·27-s − 1.77·29-s − 0.550·31-s + 0.135i·33-s − 0.838i·37-s + 0.280·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{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: $$1100$$    =    $$2^{2} \cdot 5^{2} \cdot 11$$ Sign: $-0.447 - 0.894i$ Analytic conductor: $$176.422$$ Root analytic conductor: $$13.2824$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{1100} (749, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1100,\ (\ :5/2),\ -0.447 - 0.894i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.115057685$$ $$L(\frac12)$$ $$\approx$$ $$2.115057685$$ $$L(\frac{7}{2})$$ 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$$
11 $$1 - 121T$$
good3 $$1 - 7iT - 243T^{2}$$
7 $$1 - 50iT - 1.68e4T^{2}$$
13 $$1 + 380iT - 3.71e5T^{2}$$
17 $$1 - 1.15e3iT - 1.41e6T^{2}$$
19 $$1 - 1.82e3T + 2.47e6T^{2}$$
23 $$1 - 3.59e3iT - 6.43e6T^{2}$$
29 $$1 + 8.03e3T + 2.05e7T^{2}$$
31 $$1 + 2.94e3T + 2.86e7T^{2}$$
37 $$1 + 6.97e3iT - 6.93e7T^{2}$$
41 $$1 + 520T + 1.15e8T^{2}$$
43 $$1 + 2.48e3iT - 1.47e8T^{2}$$
47 $$1 - 6.92e3iT - 2.29e8T^{2}$$
53 $$1 + 1.37e4iT - 4.18e8T^{2}$$
59 $$1 - 3.17e4T + 7.14e8T^{2}$$
61 $$1 - 3.41e4T + 8.44e8T^{2}$$
67 $$1 - 6.15e4iT - 1.35e9T^{2}$$
71 $$1 + 1.49e4T + 1.80e9T^{2}$$
73 $$1 + 3.64e4iT - 2.07e9T^{2}$$
79 $$1 - 2.85e4T + 3.07e9T^{2}$$
83 $$1 - 7.74e4iT - 3.93e9T^{2}$$
89 $$1 + 3.62e4T + 5.58e9T^{2}$$
97 $$1 - 4.97e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$