# Properties

 Degree $2$ Conductor $38$ Sign $1$ Motivic weight $9$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 16·2-s + 25.2·3-s + 256·4-s + 2.12e3·5-s − 404.·6-s + 1.14e4·7-s − 4.09e3·8-s − 1.90e4·9-s − 3.40e4·10-s − 1.04e4·11-s + 6.46e3·12-s + 1.44e5·13-s − 1.83e5·14-s + 5.37e4·15-s + 6.55e4·16-s − 6.54e5·17-s + 3.04e5·18-s + 1.30e5·19-s + 5.44e5·20-s + 2.89e5·21-s + 1.66e5·22-s + 1.63e6·23-s − 1.03e5·24-s + 2.56e6·25-s − 2.31e6·26-s − 9.78e5·27-s + 2.93e6·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.180·3-s + 0.5·4-s + 1.52·5-s − 0.127·6-s + 1.80·7-s − 0.353·8-s − 0.967·9-s − 1.07·10-s − 0.214·11-s + 0.0900·12-s + 1.40·13-s − 1.27·14-s + 0.273·15-s + 0.250·16-s − 1.89·17-s + 0.684·18-s + 0.229·19-s + 0.760·20-s + 0.325·21-s + 0.151·22-s + 1.21·23-s − 0.0636·24-s + 1.31·25-s − 0.993·26-s − 0.354·27-s + 0.902·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $1$ Motivic weight: $$9$$ Character: $\chi_{38} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :9/2),\ 1)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$2.21899$$ $$L(\frac12)$$ $$\approx$$ $$2.21899$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 16T$$
19 $$1 - 1.30e5T$$
good3 $$1 - 25.2T + 1.96e4T^{2}$$
5 $$1 - 2.12e3T + 1.95e6T^{2}$$
7 $$1 - 1.14e4T + 4.03e7T^{2}$$
11 $$1 + 1.04e4T + 2.35e9T^{2}$$
13 $$1 - 1.44e5T + 1.06e10T^{2}$$
17 $$1 + 6.54e5T + 1.18e11T^{2}$$
23 $$1 - 1.63e6T + 1.80e12T^{2}$$
29 $$1 - 3.60e6T + 1.45e13T^{2}$$
31 $$1 - 3.62e6T + 2.64e13T^{2}$$
37 $$1 + 1.06e7T + 1.29e14T^{2}$$
41 $$1 - 1.31e7T + 3.27e14T^{2}$$
43 $$1 + 2.73e6T + 5.02e14T^{2}$$
47 $$1 + 5.26e7T + 1.11e15T^{2}$$
53 $$1 - 5.87e7T + 3.29e15T^{2}$$
59 $$1 + 7.23e7T + 8.66e15T^{2}$$
61 $$1 + 7.46e7T + 1.16e16T^{2}$$
67 $$1 - 1.12e8T + 2.72e16T^{2}$$
71 $$1 - 3.11e8T + 4.58e16T^{2}$$
73 $$1 - 3.47e6T + 5.88e16T^{2}$$
79 $$1 - 1.96e8T + 1.19e17T^{2}$$
83 $$1 + 3.30e8T + 1.86e17T^{2}$$
89 $$1 + 6.40e8T + 3.50e17T^{2}$$
97 $$1 - 1.56e9T + 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$