# 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 + 3.23·3-s + 64·4-s − 312.·5-s + 25.9·6-s − 766.·7-s + 512·8-s − 2.17e3·9-s − 2.49e3·10-s + 252.·11-s + 207.·12-s − 1.06e3·13-s − 6.13e3·14-s − 1.01e3·15-s + 4.09e3·16-s − 1.87e4·17-s − 1.74e4·18-s − 6.85e3·19-s − 1.99e4·20-s − 2.48e3·21-s + 2.01e3·22-s + 1.15e4·23-s + 1.65e3·24-s + 1.95e4·25-s − 8.52e3·26-s − 1.41e4·27-s − 4.90e4·28-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.0692·3-s + 0.5·4-s − 1.11·5-s + 0.0489·6-s − 0.844·7-s + 0.353·8-s − 0.995·9-s − 0.790·10-s + 0.0570·11-s + 0.0346·12-s − 0.134·13-s − 0.597·14-s − 0.0774·15-s + 0.250·16-s − 0.926·17-s − 0.703·18-s − 0.229·19-s − 0.558·20-s − 0.0585·21-s + 0.0403·22-s + 0.198·23-s + 0.0244·24-s + 0.249·25-s − 0.0950·26-s − 0.138·27-s − 0.422·28-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 - 3.23T + 2.18e3T^{2}$$
5 $$1 + 312.T + 7.81e4T^{2}$$
7 $$1 + 766.T + 8.23e5T^{2}$$
11 $$1 - 252.T + 1.94e7T^{2}$$
13 $$1 + 1.06e3T + 6.27e7T^{2}$$
17 $$1 + 1.87e4T + 4.10e8T^{2}$$
23 $$1 - 1.15e4T + 3.40e9T^{2}$$
29 $$1 + 4.62e4T + 1.72e10T^{2}$$
31 $$1 - 4.68e4T + 2.75e10T^{2}$$
37 $$1 + 1.82e5T + 9.49e10T^{2}$$
41 $$1 - 8.19e5T + 1.94e11T^{2}$$
43 $$1 - 4.77e5T + 2.71e11T^{2}$$
47 $$1 - 9.92e5T + 5.06e11T^{2}$$
53 $$1 + 8.52e5T + 1.17e12T^{2}$$
59 $$1 + 1.92e6T + 2.48e12T^{2}$$
61 $$1 - 2.09e5T + 3.14e12T^{2}$$
67 $$1 + 2.32e6T + 6.06e12T^{2}$$
71 $$1 + 5.37e6T + 9.09e12T^{2}$$
73 $$1 + 3.71e6T + 1.10e13T^{2}$$
79 $$1 + 1.30e6T + 1.92e13T^{2}$$
83 $$1 + 6.51e6T + 2.71e13T^{2}$$
89 $$1 - 3.99e6T + 4.42e13T^{2}$$
97 $$1 + 6.90e6T + 8.07e13T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$