# Properties

 Degree $2$ Conductor $38$ Sign $0.976 + 0.216i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.82i·2-s + 6.66i·3-s − 8.00·4-s + 26.7·5-s + 18.8·6-s + 51.4·7-s + 22.6i·8-s + 36.5·9-s − 75.5i·10-s − 25.0·11-s − 53.3i·12-s − 53.2i·13-s − 145. i·14-s + 177. i·15-s + 64.0·16-s − 24.4·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s + 0.740i·3-s − 0.500·4-s + 1.06·5-s + 0.523·6-s + 1.04·7-s + 0.353i·8-s + 0.451·9-s − 0.755i·10-s − 0.207·11-s − 0.370i·12-s − 0.315i·13-s − 0.742i·14-s + 0.790i·15-s + 0.250·16-s − 0.0847·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $0.976 + 0.216i$ Motivic weight: $$4$$ Character: $\chi_{38} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :2),\ 0.976 + 0.216i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.71647 - 0.187677i$$ $$L(\frac12)$$ $$\approx$$ $$1.71647 - 0.187677i$$ $$L(3)$$ 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 + (78.0 - 352. i)T$$
good3 $$1 - 6.66iT - 81T^{2}$$
5 $$1 - 26.7T + 625T^{2}$$
7 $$1 - 51.4T + 2.40e3T^{2}$$
11 $$1 + 25.0T + 1.46e4T^{2}$$
13 $$1 + 53.2iT - 2.85e4T^{2}$$
17 $$1 + 24.4T + 8.35e4T^{2}$$
23 $$1 + 612.T + 2.79e5T^{2}$$
29 $$1 + 1.34e3iT - 7.07e5T^{2}$$
31 $$1 + 912. iT - 9.23e5T^{2}$$
37 $$1 - 325. iT - 1.87e6T^{2}$$
41 $$1 + 51.7iT - 2.82e6T^{2}$$
43 $$1 + 2.54e3T + 3.41e6T^{2}$$
47 $$1 - 2.99e3T + 4.87e6T^{2}$$
53 $$1 + 4.06e3iT - 7.89e6T^{2}$$
59 $$1 - 1.07e3iT - 1.21e7T^{2}$$
61 $$1 + 6.53e3T + 1.38e7T^{2}$$
67 $$1 - 7.38e3iT - 2.01e7T^{2}$$
71 $$1 - 5.80e3iT - 2.54e7T^{2}$$
73 $$1 - 3.62e3T + 2.83e7T^{2}$$
79 $$1 - 3.90e3iT - 3.89e7T^{2}$$
83 $$1 + 997.T + 4.74e7T^{2}$$
89 $$1 + 6.15e3iT - 6.27e7T^{2}$$
97 $$1 - 5.02e3iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$