# Properties

 Label 2-3800-152.37-c0-0-18 Degree $2$ Conductor $3800$ Sign $1$ Analytic cond. $1.89644$ Root an. cond. $1.37711$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 0.347·3-s + 4-s + 0.347·6-s + 1.87·7-s + 8-s − 0.879·9-s + 0.347·12-s + 1.53·13-s + 1.87·14-s + 16-s − 1.53·17-s − 0.879·18-s − 19-s + 0.652·21-s − 0.347·23-s + 0.347·24-s + 1.53·26-s − 0.652·27-s + 1.87·28-s − 1.53·29-s + 32-s − 1.53·34-s − 0.879·36-s − 37-s − 38-s + 0.532·39-s + ⋯
 L(s)  = 1 + 2-s + 0.347·3-s + 4-s + 0.347·6-s + 1.87·7-s + 8-s − 0.879·9-s + 0.347·12-s + 1.53·13-s + 1.87·14-s + 16-s − 1.53·17-s − 0.879·18-s − 19-s + 0.652·21-s − 0.347·23-s + 0.347·24-s + 1.53·26-s − 0.652·27-s + 1.87·28-s − 1.53·29-s + 32-s − 1.53·34-s − 0.879·36-s − 37-s − 38-s + 0.532·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3800$$    =    $$2^{3} \cdot 5^{2} \cdot 19$$ Sign: $1$ Analytic conductor: $$1.89644$$ Root analytic conductor: $$1.37711$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3800} (1101, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 3800,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$3.194123755$$ $$L(\frac12)$$ $$\approx$$ $$3.194123755$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
5 $$1$$
19 $$1 + T$$
good3 $$1 - 0.347T + T^{2}$$
7 $$1 - 1.87T + T^{2}$$
11 $$1 - T^{2}$$
13 $$1 - 1.53T + T^{2}$$
17 $$1 + 1.53T + T^{2}$$
23 $$1 + 0.347T + T^{2}$$
29 $$1 + 1.53T + T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + T + T^{2}$$
41 $$1 - T^{2}$$
43 $$1 - T^{2}$$
47 $$1 - T + T^{2}$$
53 $$1 + 1.87T + T^{2}$$
59 $$1 + 0.347T + T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + 1.87T + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + 0.347T + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$