# Properties

 Label 2-147-21.20-c5-0-32 Degree $2$ Conductor $147$ Sign $0.791 - 0.610i$ Analytic cond. $23.5764$ Root an. cond. $4.85555$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.65i·2-s + (−15.5 − 0.882i)3-s + 18.6·4-s + 74.3·5-s + (3.22 − 56.8i)6-s + 185. i·8-s + (241. + 27.4i)9-s + 271. i·10-s − 39.3i·11-s + (−290. − 16.4i)12-s − 589. i·13-s + (−1.15e3 − 65.5i)15-s − 80.0·16-s + 1.23e3·17-s + (−100. + 882. i)18-s − 2.76e3i·19-s + ⋯
 L(s)  = 1 + 0.646i·2-s + (−0.998 − 0.0566i)3-s + 0.582·4-s + 1.32·5-s + (0.0365 − 0.645i)6-s + 1.02i·8-s + (0.993 + 0.113i)9-s + 0.859i·10-s − 0.0981i·11-s + (−0.581 − 0.0329i)12-s − 0.967i·13-s + (−1.32 − 0.0752i)15-s − 0.0781·16-s + 1.03·17-s + (−0.0730 + 0.641i)18-s − 1.75i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.791 - 0.610i$ Analytic conductor: $$23.5764$$ Root analytic conductor: $$4.85555$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{147} (146, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :5/2),\ 0.791 - 0.610i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.278875503$$ $$L(\frac12)$$ $$\approx$$ $$2.278875503$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (15.5 + 0.882i)T$$
7 $$1$$
good2 $$1 - 3.65iT - 32T^{2}$$
5 $$1 - 74.3T + 3.12e3T^{2}$$
11 $$1 + 39.3iT - 1.61e5T^{2}$$
13 $$1 + 589. iT - 3.71e5T^{2}$$
17 $$1 - 1.23e3T + 1.41e6T^{2}$$
19 $$1 + 2.76e3iT - 2.47e6T^{2}$$
23 $$1 + 463. iT - 6.43e6T^{2}$$
29 $$1 - 5.29e3iT - 2.05e7T^{2}$$
31 $$1 - 2.84e3iT - 2.86e7T^{2}$$
37 $$1 - 7.84e3T + 6.93e7T^{2}$$
41 $$1 - 1.21e4T + 1.15e8T^{2}$$
43 $$1 - 5.35e3T + 1.47e8T^{2}$$
47 $$1 - 6.75e3T + 2.29e8T^{2}$$
53 $$1 - 1.60e4iT - 4.18e8T^{2}$$
59 $$1 - 7.74e3T + 7.14e8T^{2}$$
61 $$1 + 5.61e3iT - 8.44e8T^{2}$$
67 $$1 + 1.28e4T + 1.35e9T^{2}$$
71 $$1 + 6.06e4iT - 1.80e9T^{2}$$
73 $$1 - 5.16e4iT - 2.07e9T^{2}$$
79 $$1 - 6.73e4T + 3.07e9T^{2}$$
83 $$1 + 8.71e3T + 3.93e9T^{2}$$
89 $$1 + 1.81e4T + 5.58e9T^{2}$$
97 $$1 - 1.86e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$