# Properties

 Degree $2$ Conductor $100$ Sign $0.447 + 0.894i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 22i·3-s − 218i·7-s − 241·9-s − 480·11-s − 622i·13-s − 186i·17-s + 1.20e3·19-s + 4.79e3·21-s − 3.18e3i·23-s + 44i·27-s − 5.52e3·29-s + 9.35e3·31-s − 1.05e4i·33-s − 5.61e3i·37-s + 1.36e4·39-s + ⋯
 L(s)  = 1 + 1.41i·3-s − 1.68i·7-s − 0.991·9-s − 1.19·11-s − 1.02i·13-s − 0.156i·17-s + 0.765·19-s + 2.37·21-s − 1.25i·23-s + 0.0116i·27-s − 1.22·29-s + 1.74·31-s − 1.68i·33-s − 0.674i·37-s + 1.44·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{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: $$100$$    =    $$2^{2} \cdot 5^{2}$$ Sign: $0.447 + 0.894i$ Motivic weight: $$5$$ Character: $\chi_{100} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 100,\ (\ :5/2),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.948069 - 0.585938i$$ $$L(\frac12)$$ $$\approx$$ $$0.948069 - 0.585938i$$ $$L(\frac{7}{2})$$ 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 - 22iT - 243T^{2}$$
7 $$1 + 218iT - 1.68e4T^{2}$$
11 $$1 + 480T + 1.61e5T^{2}$$
13 $$1 + 622iT - 3.71e5T^{2}$$
17 $$1 + 186iT - 1.41e6T^{2}$$
19 $$1 - 1.20e3T + 2.47e6T^{2}$$
23 $$1 + 3.18e3iT - 6.43e6T^{2}$$
29 $$1 + 5.52e3T + 2.05e7T^{2}$$
31 $$1 - 9.35e3T + 2.86e7T^{2}$$
37 $$1 + 5.61e3iT - 6.93e7T^{2}$$
41 $$1 + 1.43e4T + 1.15e8T^{2}$$
43 $$1 + 370iT - 1.47e8T^{2}$$
47 $$1 + 1.61e4iT - 2.29e8T^{2}$$
53 $$1 + 4.37e3iT - 4.18e8T^{2}$$
59 $$1 - 1.17e4T + 7.14e8T^{2}$$
61 $$1 - 1.32e4T + 8.44e8T^{2}$$
67 $$1 - 1.15e4iT - 1.35e9T^{2}$$
71 $$1 + 2.95e4T + 1.80e9T^{2}$$
73 $$1 - 3.36e4iT - 2.07e9T^{2}$$
79 $$1 + 3.12e4T + 3.07e9T^{2}$$
83 $$1 + 3.84e4iT - 3.93e9T^{2}$$
89 $$1 + 1.19e5T + 5.58e9T^{2}$$
97 $$1 + 9.46e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$