# Properties

 Label 1-760-760.189-r1-0-0 Degree $1$ Conductor $760$ Sign $1$ Analytic cond. $81.6733$ Root an. cond. $81.6733$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s − 7-s + 9-s − 11-s − 13-s − 17-s + 21-s − 23-s − 27-s + 29-s − 31-s + 33-s − 37-s + 39-s − 41-s + 43-s − 47-s + 49-s + 51-s − 53-s + 59-s − 61-s − 63-s − 67-s + 69-s − 71-s − 73-s + ⋯
 L(s)  = 1 − 3-s − 7-s + 9-s − 11-s − 13-s − 17-s + 21-s − 23-s − 27-s + 29-s − 31-s + 33-s − 37-s + 39-s − 41-s + 43-s − 47-s + 49-s + 51-s − 53-s + 59-s − 61-s − 63-s − 67-s + 69-s − 71-s − 73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$760$$    =    $$2^{3} \cdot 5 \cdot 19$$ Sign: $1$ Analytic conductor: $$81.6733$$ Root analytic conductor: $$81.6733$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{760} (189, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 760,\ (1:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1940667160$$ $$L(\frac12)$$ $$\approx$$ $$0.1940667160$$ $$L(1)$$ $$\approx$$ $$0.4558301715$$ $$L(1)$$ $$\approx$$ $$0.4558301715$$

## Euler product

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