# Properties

 Label 1-304-304.37-r1-0-0 Degree $1$ Conductor $304$ Sign $-0.382 - 0.923i$ Analytic cond. $32.6693$ Root an. cond. $32.6693$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + i·5-s − 7-s − 9-s + i·11-s + i·13-s − 15-s + 17-s − i·21-s − 23-s − 25-s − i·27-s + i·29-s − 31-s − 33-s + ⋯
 L(s)  = 1 + i·3-s + i·5-s − 7-s − 9-s + i·11-s + i·13-s − 15-s + 17-s − i·21-s − 23-s − 25-s − i·27-s + i·29-s − 31-s − 33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$304$$    =    $$2^{4} \cdot 19$$ Sign: $-0.382 - 0.923i$ Analytic conductor: $$32.6693$$ Root analytic conductor: $$32.6693$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{304} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 304,\ (1:\ ),\ -0.382 - 0.923i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$-0.3721904942 + 0.5570224384i$$ $$L(\frac12)$$ $$\approx$$ $$-0.3721904942 + 0.5570224384i$$ $$L(1)$$ $$\approx$$ $$0.6087459402 + 0.5279625547i$$ $$L(1)$$ $$\approx$$ $$0.6087459402 + 0.5279625547i$$

## Euler product

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