# Properties

 Label 2-147-21.5-c3-0-8 Degree $2$ Conductor $147$ Sign $0.444 - 0.895i$ Analytic cond. $8.67328$ Root an. cond. $2.94504$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.5 + 2.59i)3-s + (−4 − 6.92i)4-s + (13.5 − 23.3i)9-s + (36 + 20.7i)12-s + 62.3i·13-s + (−31.9 + 55.4i)16-s + (135 + 77.9i)19-s + (62.5 + 108. i)25-s + 140. i·27-s + (−135 + 77.9i)31-s − 216·36-s + (55 − 95.2i)37-s + (−162 − 280. i)39-s + 520·43-s − 332. i·48-s + ⋯
 L(s)  = 1 + (−0.866 + 0.499i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)9-s + (0.866 + 0.499i)12-s + 1.33i·13-s + (−0.499 + 0.866i)16-s + (1.63 + 0.941i)19-s + (0.5 + 0.866i)25-s + 1.00i·27-s + (−0.782 + 0.451i)31-s − 36-s + (0.244 − 0.423i)37-s + (−0.665 − 1.15i)39-s + 1.84·43-s − 0.999i·48-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.444 - 0.895i$ Analytic conductor: $$8.67328$$ Root analytic conductor: $$2.94504$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{147} (68, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :3/2),\ 0.444 - 0.895i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.775719 + 0.481258i$$ $$L(\frac12)$$ $$\approx$$ $$0.775719 + 0.481258i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (4.5 - 2.59i)T$$
7 $$1$$
good2 $$1 + (4 + 6.92i)T^{2}$$
5 $$1 + (-62.5 - 108. i)T^{2}$$
11 $$1 + (665.5 - 1.15e3i)T^{2}$$
13 $$1 - 62.3iT - 2.19e3T^{2}$$
17 $$1 + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-135 - 77.9i)T + (3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (6.08e3 + 1.05e4i)T^{2}$$
29 $$1 - 2.43e4T^{2}$$
31 $$1 + (135 - 77.9i)T + (1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (-55 + 95.2i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 - 520T + 7.95e4T^{2}$$
47 $$1 + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (810 + 467. i)T + (1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-440 - 762. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 3.57e5T^{2}$$
73 $$1 + (324 - 187. i)T + (1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (442 - 765. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 - 1.37e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$