# Properties

 Degree $2$ Conductor $38$ Sign $-0.972 + 0.234i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−3.75 − 1.36i)2-s + (−5.14 − 29.1i)3-s + (12.2 + 10.2i)4-s + (68.5 − 57.5i)5-s + (−20.5 + 116. i)6-s + (68.7 − 119. i)7-s + (−32.0 − 55.4i)8-s + (−597. + 217. i)9-s + (−336. + 122. i)10-s + (152. + 264. i)11-s + (237. − 410. i)12-s + (15.9 − 90.4i)13-s + (−421. + 353. i)14-s + (−2.03e3 − 1.70e3i)15-s + (44.4 + 252. i)16-s + (54.3 + 19.7i)17-s + ⋯
 L(s)  = 1 + (−0.664 − 0.241i)2-s + (−0.330 − 1.87i)3-s + (0.383 + 0.321i)4-s + (1.22 − 1.02i)5-s + (−0.233 + 1.32i)6-s + (0.529 − 0.917i)7-s + (−0.176 − 0.306i)8-s + (−2.46 + 0.895i)9-s + (−1.06 + 0.387i)10-s + (0.380 + 0.658i)11-s + (0.475 − 0.823i)12-s + (0.0261 − 0.148i)13-s + (−0.574 + 0.481i)14-s + (−2.33 − 1.95i)15-s + (0.0434 + 0.246i)16-s + (0.0455 + 0.0165i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $-0.972 + 0.234i$ Motivic weight: $$5$$ Character: $\chi_{38} (23, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :5/2),\ -0.972 + 0.234i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.147199 - 1.23708i$$ $$L(\frac12)$$ $$\approx$$ $$0.147199 - 1.23708i$$ $$L(\frac{7}{2})$$ 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 + (-920. - 1.27e3i)T$$
good3 $$1 + (5.14 + 29.1i)T + (-228. + 83.1i)T^{2}$$
5 $$1 + (-68.5 + 57.5i)T + (542. - 3.07e3i)T^{2}$$
7 $$1 + (-68.7 + 119. i)T + (-8.40e3 - 1.45e4i)T^{2}$$
11 $$1 + (-152. - 264. i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 + (-15.9 + 90.4i)T + (-3.48e5 - 1.26e5i)T^{2}$$
17 $$1 + (-54.3 - 19.7i)T + (1.08e6 + 9.12e5i)T^{2}$$
23 $$1 + (-2.22e3 - 1.86e3i)T + (1.11e6 + 6.33e6i)T^{2}$$
29 $$1 + (5.07e3 - 1.84e3i)T + (1.57e7 - 1.31e7i)T^{2}$$
31 $$1 + (-2.96e3 + 5.14e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + 287.T + 6.93e7T^{2}$$
41 $$1 + (240. + 1.36e3i)T + (-1.08e8 + 3.96e7i)T^{2}$$
43 $$1 + (-6.18e3 + 5.18e3i)T + (2.55e7 - 1.44e8i)T^{2}$$
47 $$1 + (1.47e3 - 537. i)T + (1.75e8 - 1.47e8i)T^{2}$$
53 $$1 + (2.96e4 + 2.49e4i)T + (7.26e7 + 4.11e8i)T^{2}$$
59 $$1 + (1.90e3 + 692. i)T + (5.47e8 + 4.59e8i)T^{2}$$
61 $$1 + (-1.66e4 - 1.40e4i)T + (1.46e8 + 8.31e8i)T^{2}$$
67 $$1 + (-5.93e4 + 2.15e4i)T + (1.03e9 - 8.67e8i)T^{2}$$
71 $$1 + (1.92e4 - 1.61e4i)T + (3.13e8 - 1.77e9i)T^{2}$$
73 $$1 + (-5.90e3 - 3.34e4i)T + (-1.94e9 + 7.09e8i)T^{2}$$
79 $$1 + (2.42e3 + 1.37e4i)T + (-2.89e9 + 1.05e9i)T^{2}$$
83 $$1 + (4.01e4 - 6.94e4i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 + (8.64e3 - 4.90e4i)T + (-5.24e9 - 1.90e9i)T^{2}$$
97 $$1 + (3.82e4 + 1.39e4i)T + (6.57e9 + 5.51e9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$