# Properties

 Degree $2$ Conductor $38$ Sign $-1$ Motivic weight $7$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 8·2-s − 72.2·3-s + 64·4-s + 467.·5-s − 577.·6-s − 1.47e3·7-s + 512·8-s + 3.03e3·9-s + 3.73e3·10-s − 3.54e3·11-s − 4.62e3·12-s − 1.23e4·13-s − 1.17e4·14-s − 3.37e4·15-s + 4.09e3·16-s − 1.34e4·17-s + 2.42e4·18-s − 6.85e3·19-s + 2.99e4·20-s + 1.06e5·21-s − 2.83e4·22-s − 9.40e4·23-s − 3.69e4·24-s + 1.40e5·25-s − 9.88e4·26-s − 6.10e4·27-s − 9.41e4·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1.54·3-s + 0.5·4-s + 1.67·5-s − 1.09·6-s − 1.62·7-s + 0.353·8-s + 1.38·9-s + 1.18·10-s − 0.803·11-s − 0.772·12-s − 1.56·13-s − 1.14·14-s − 2.58·15-s + 0.250·16-s − 0.665·17-s + 0.980·18-s − 0.229·19-s + 0.836·20-s + 2.50·21-s − 0.568·22-s − 1.61·23-s − 0.546·24-s + 1.79·25-s − 1.10·26-s − 0.596·27-s − 0.810·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $-1$ Motivic weight: $$7$$ Character: $\chi_{38} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 38,\ (\ :7/2),\ -1)$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 8T$$
19 $$1 + 6.85e3T$$
good3 $$1 + 72.2T + 2.18e3T^{2}$$
5 $$1 - 467.T + 7.81e4T^{2}$$
7 $$1 + 1.47e3T + 8.23e5T^{2}$$
11 $$1 + 3.54e3T + 1.94e7T^{2}$$
13 $$1 + 1.23e4T + 6.27e7T^{2}$$
17 $$1 + 1.34e4T + 4.10e8T^{2}$$
23 $$1 + 9.40e4T + 3.40e9T^{2}$$
29 $$1 - 3.35e4T + 1.72e10T^{2}$$
31 $$1 - 2.12e5T + 2.75e10T^{2}$$
37 $$1 - 3.36e4T + 9.49e10T^{2}$$
41 $$1 + 4.80e5T + 1.94e11T^{2}$$
43 $$1 + 5.61e5T + 2.71e11T^{2}$$
47 $$1 - 4.78e5T + 5.06e11T^{2}$$
53 $$1 + 9.31e4T + 1.17e12T^{2}$$
59 $$1 - 9.55e5T + 2.48e12T^{2}$$
61 $$1 + 1.71e6T + 3.14e12T^{2}$$
67 $$1 - 4.76e5T + 6.06e12T^{2}$$
71 $$1 - 1.95e6T + 9.09e12T^{2}$$
73 $$1 - 1.21e6T + 1.10e13T^{2}$$
79 $$1 - 3.94e6T + 1.92e13T^{2}$$
83 $$1 + 3.54e6T + 2.71e13T^{2}$$
89 $$1 + 7.50e6T + 4.42e13T^{2}$$
97 $$1 - 1.27e7T + 8.07e13T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$