# Properties

 Degree $2$ Conductor $38$ Sign $-0.999 + 0.0387i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.82i·2-s + 7.80i·3-s − 8.00·4-s − 33.0·5-s − 22.0·6-s + 16.0·7-s − 22.6i·8-s + 20.1·9-s − 93.6i·10-s − 215.·11-s − 62.4i·12-s + 281. i·13-s + 45.4i·14-s − 258. i·15-s + 64.0·16-s + 226.·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + 0.867i·3-s − 0.500·4-s − 1.32·5-s − 0.613·6-s + 0.328·7-s − 0.353i·8-s + 0.248·9-s − 0.936i·10-s − 1.78·11-s − 0.433i·12-s + 1.66i·13-s + 0.232i·14-s − 1.14i·15-s + 0.250·16-s + 0.784·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0387i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $-0.999 + 0.0387i$ Motivic weight: $$4$$ Character: $\chi_{38} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :2),\ -0.999 + 0.0387i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.0155537 - 0.802003i$$ $$L(\frac12)$$ $$\approx$$ $$0.0155537 - 0.802003i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 2.82iT$$
19 $$1 + (-13.9 - 360. i)T$$
good3 $$1 - 7.80iT - 81T^{2}$$
5 $$1 + 33.0T + 625T^{2}$$
7 $$1 - 16.0T + 2.40e3T^{2}$$
11 $$1 + 215.T + 1.46e4T^{2}$$
13 $$1 - 281. iT - 2.85e4T^{2}$$
17 $$1 - 226.T + 8.35e4T^{2}$$
23 $$1 - 414.T + 2.79e5T^{2}$$
29 $$1 + 606. iT - 7.07e5T^{2}$$
31 $$1 - 478. iT - 9.23e5T^{2}$$
37 $$1 - 104. iT - 1.87e6T^{2}$$
41 $$1 + 1.89e3iT - 2.82e6T^{2}$$
43 $$1 - 315.T + 3.41e6T^{2}$$
47 $$1 + 474.T + 4.87e6T^{2}$$
53 $$1 - 774. iT - 7.89e6T^{2}$$
59 $$1 + 567. iT - 1.21e7T^{2}$$
61 $$1 - 4.79e3T + 1.38e7T^{2}$$
67 $$1 + 4.46e3iT - 2.01e7T^{2}$$
71 $$1 - 7.63e3iT - 2.54e7T^{2}$$
73 $$1 - 8.39e3T + 2.83e7T^{2}$$
79 $$1 - 9.70e3iT - 3.89e7T^{2}$$
83 $$1 + 1.05e4T + 4.74e7T^{2}$$
89 $$1 - 1.02e4iT - 6.27e7T^{2}$$
97 $$1 + 1.54e4iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$