# Properties

 Label 1-304-304.11-r1-0-0 Degree $1$ Conductor $304$ Sign $0.337 - 0.941i$ 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 + (0.866 − 0.5i)3-s + (0.866 − 0.5i)5-s + 7-s + (0.5 − 0.866i)9-s − i·11-s + (0.866 + 0.5i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)21-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s + (0.866 + 0.5i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯
 L(s)  = 1 + (0.866 − 0.5i)3-s + (0.866 − 0.5i)5-s + 7-s + (0.5 − 0.866i)9-s − i·11-s + (0.866 + 0.5i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)21-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s + (0.866 + 0.5i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$304$$    =    $$2^{4} \cdot 19$$ Sign: $0.337 - 0.941i$ Analytic conductor: $$32.6693$$ Root analytic conductor: $$32.6693$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{304} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 304,\ (1:\ ),\ 0.337 - 0.941i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$3.029784020 - 2.133494604i$$ $$L(\frac12)$$ $$\approx$$ $$3.029784020 - 2.133494604i$$ $$L(1)$$ $$\approx$$ $$1.799214924 - 0.6443174081i$$ $$L(1)$$ $$\approx$$ $$1.799214924 - 0.6443174081i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1$$
good3 $$1 + (0.866 - 0.5i)T$$
5 $$1 + (0.866 - 0.5i)T$$
7 $$1 + T$$
11 $$1 - iT$$
13 $$1 + (0.866 + 0.5i)T$$
17 $$1 + (-0.5 - 0.866i)T$$
23 $$1 + (-0.5 + 0.866i)T$$
29 $$1 + (0.866 + 0.5i)T$$
31 $$1 - T$$
37 $$1 + iT$$
41 $$1 + (0.5 + 0.866i)T$$
43 $$1 + (-0.866 + 0.5i)T$$
47 $$1 + (0.5 - 0.866i)T$$
53 $$1 + (-0.866 - 0.5i)T$$
59 $$1 + (-0.866 + 0.5i)T$$
61 $$1 + (0.866 + 0.5i)T$$
67 $$1 + (-0.866 - 0.5i)T$$
71 $$1 + (-0.5 - 0.866i)T$$
73 $$1 + (0.5 + 0.866i)T$$
79 $$1 + (0.5 + 0.866i)T$$
83 $$1 + iT$$
89 $$1 + (0.5 - 0.866i)T$$
97 $$1 + (-0.5 - 0.866i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$