# Properties

 Degree $2$ Conductor $38$ Sign $1$ Motivic weight $5$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4·2-s + 20.4·3-s + 16·4-s + 34.4·5-s − 81.9·6-s − 18.9·7-s − 64·8-s + 176.·9-s − 137.·10-s + 349.·11-s + 327.·12-s + 711.·13-s + 75.6·14-s + 705.·15-s + 256·16-s + 221.·17-s − 705.·18-s + 361·19-s + 551.·20-s − 387.·21-s − 1.39e3·22-s − 662.·23-s − 1.31e3·24-s − 1.93e3·25-s − 2.84e3·26-s − 1.36e3·27-s − 302.·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1.31·3-s + 0.5·4-s + 0.616·5-s − 0.929·6-s − 0.145·7-s − 0.353·8-s + 0.726·9-s − 0.435·10-s + 0.870·11-s + 0.656·12-s + 1.16·13-s + 0.103·14-s + 0.809·15-s + 0.250·16-s + 0.186·17-s − 0.513·18-s + 0.229·19-s + 0.308·20-s − 0.191·21-s − 0.615·22-s − 0.261·23-s − 0.464·24-s − 0.620·25-s − 0.825·26-s − 0.359·27-s − 0.0729·28-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $1$ Motivic weight: $$5$$ Character: $\chi_{38} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.91996$$ $$L(\frac12)$$ $$\approx$$ $$1.91996$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 4T$$
19 $$1 - 361T$$
good3 $$1 - 20.4T + 243T^{2}$$
5 $$1 - 34.4T + 3.12e3T^{2}$$
7 $$1 + 18.9T + 1.68e4T^{2}$$
11 $$1 - 349.T + 1.61e5T^{2}$$
13 $$1 - 711.T + 3.71e5T^{2}$$
17 $$1 - 221.T + 1.41e6T^{2}$$
23 $$1 + 662.T + 6.43e6T^{2}$$
29 $$1 + 7.21e3T + 2.05e7T^{2}$$
31 $$1 - 5.40e3T + 2.86e7T^{2}$$
37 $$1 - 1.97e3T + 6.93e7T^{2}$$
41 $$1 + 3.11e3T + 1.15e8T^{2}$$
43 $$1 - 318.T + 1.47e8T^{2}$$
47 $$1 + 2.72e4T + 2.29e8T^{2}$$
53 $$1 + 1.11e3T + 4.18e8T^{2}$$
59 $$1 + 3.79e4T + 7.14e8T^{2}$$
61 $$1 - 3.74e4T + 8.44e8T^{2}$$
67 $$1 + 5.49e4T + 1.35e9T^{2}$$
71 $$1 + 7.17e3T + 1.80e9T^{2}$$
73 $$1 - 6.47e4T + 2.07e9T^{2}$$
79 $$1 - 3.61e4T + 3.07e9T^{2}$$
83 $$1 + 5.17e4T + 3.93e9T^{2}$$
89 $$1 - 1.45e5T + 5.58e9T^{2}$$
97 $$1 - 3.95e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$