# Properties

 Degree $2$ Conductor $38$ Sign $-0.990 - 0.134i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−3.75 + 1.36i)2-s + (4.41 − 25.0i)3-s + (12.2 − 10.2i)4-s + (−71.5 − 60.0i)5-s + (17.6 + 100. i)6-s + (91.0 + 157. i)7-s + (−32.0 + 55.4i)8-s + (−377. − 137. i)9-s + (351. + 127. i)10-s + (−198. + 344. i)11-s + (−203. − 351. i)12-s + (−119. − 680. i)13-s + (−558. − 468. i)14-s + (−1.81e3 + 1.52e3i)15-s + (44.4 − 252. i)16-s + (−1.15e3 + 421. i)17-s + ⋯
 L(s)  = 1 + (−0.664 + 0.241i)2-s + (0.282 − 1.60i)3-s + (0.383 − 0.321i)4-s + (−1.28 − 1.07i)5-s + (0.200 + 1.13i)6-s + (0.702 + 1.21i)7-s + (−0.176 + 0.306i)8-s + (−1.55 − 0.565i)9-s + (1.11 + 0.404i)10-s + (−0.495 + 0.858i)11-s + (−0.407 − 0.705i)12-s + (−0.196 − 1.11i)13-s + (−0.761 − 0.638i)14-s + (−2.08 + 1.75i)15-s + (0.0434 − 0.246i)16-s + (−0.972 + 0.353i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $-0.990 - 0.134i$ Motivic weight: $$5$$ Character: $\chi_{38} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :5/2),\ -0.990 - 0.134i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.0369392 + 0.547817i$$ $$L(\frac12)$$ $$\approx$$ $$0.0369392 + 0.547817i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (3.75 - 1.36i)T$$
19 $$1 + (-785. + 1.36e3i)T$$
good3 $$1 + (-4.41 + 25.0i)T + (-228. - 83.1i)T^{2}$$
5 $$1 + (71.5 + 60.0i)T + (542. + 3.07e3i)T^{2}$$
7 $$1 + (-91.0 - 157. i)T + (-8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + (198. - 344. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + (119. + 680. i)T + (-3.48e5 + 1.26e5i)T^{2}$$
17 $$1 + (1.15e3 - 421. i)T + (1.08e6 - 9.12e5i)T^{2}$$
23 $$1 + (-216. + 181. i)T + (1.11e6 - 6.33e6i)T^{2}$$
29 $$1 + (4.07e3 + 1.48e3i)T + (1.57e7 + 1.31e7i)T^{2}$$
31 $$1 + (1.34e3 + 2.32e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + 3.85e3T + 6.93e7T^{2}$$
41 $$1 + (-3.39e3 + 1.92e4i)T + (-1.08e8 - 3.96e7i)T^{2}$$
43 $$1 + (284. + 238. i)T + (2.55e7 + 1.44e8i)T^{2}$$
47 $$1 + (1.27e4 + 4.62e3i)T + (1.75e8 + 1.47e8i)T^{2}$$
53 $$1 + (-4.30e3 + 3.61e3i)T + (7.26e7 - 4.11e8i)T^{2}$$
59 $$1 + (-2.82e4 + 1.02e4i)T + (5.47e8 - 4.59e8i)T^{2}$$
61 $$1 + (-1.69e3 + 1.42e3i)T + (1.46e8 - 8.31e8i)T^{2}$$
67 $$1 + (249. + 90.9i)T + (1.03e9 + 8.67e8i)T^{2}$$
71 $$1 + (-1.70e4 - 1.43e4i)T + (3.13e8 + 1.77e9i)T^{2}$$
73 $$1 + (-9.67e3 + 5.48e4i)T + (-1.94e9 - 7.09e8i)T^{2}$$
79 $$1 + (9.82e3 - 5.56e4i)T + (-2.89e9 - 1.05e9i)T^{2}$$
83 $$1 + (4.25e4 + 7.37e4i)T + (-1.96e9 + 3.41e9i)T^{2}$$
89 $$1 + (-3.76e3 - 2.13e4i)T + (-5.24e9 + 1.90e9i)T^{2}$$
97 $$1 + (1.04e5 - 3.81e4i)T + (6.57e9 - 5.51e9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$