# Properties

 Label 2-147-21.20-c5-0-45 Degree $2$ Conductor $147$ Sign $-0.997 + 0.0755i$ 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 − 10.3i·2-s + (12.5 + 9.28i)3-s − 74.8·4-s + 31.7·5-s + (95.9 − 129. i)6-s + 442. i·8-s + (70.5 + 232. i)9-s − 328. i·10-s − 453. i·11-s + (−936. − 694. i)12-s − 551. i·13-s + (397. + 294. i)15-s + 2.17e3·16-s + 538.·17-s + (2.40e3 − 729. i)18-s − 1.36e3i·19-s + ⋯
 L(s)  = 1 − 1.82i·2-s + (0.803 + 0.595i)3-s − 2.33·4-s + 0.567·5-s + (1.08 − 1.46i)6-s + 2.44i·8-s + (0.290 + 0.956i)9-s − 1.03i·10-s − 1.13i·11-s + (−1.87 − 1.39i)12-s − 0.905i·13-s + (0.456 + 0.338i)15-s + 2.12·16-s + 0.451·17-s + (1.74 − 0.530i)18-s − 0.868i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0755i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $-0.997 + 0.0755i$ 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: $$(2,\ 147,\ (\ :5/2),\ -0.997 + 0.0755i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.065256194$$ $$L(\frac12)$$ $$\approx$$ $$2.065256194$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-12.5 - 9.28i)T$$
7 $$1$$
good2 $$1 + 10.3iT - 32T^{2}$$
5 $$1 - 31.7T + 3.12e3T^{2}$$
11 $$1 + 453. iT - 1.61e5T^{2}$$
13 $$1 + 551. iT - 3.71e5T^{2}$$
17 $$1 - 538.T + 1.41e6T^{2}$$
19 $$1 + 1.36e3iT - 2.47e6T^{2}$$
23 $$1 + 3.23e3iT - 6.43e6T^{2}$$
29 $$1 + 1.60e3iT - 2.05e7T^{2}$$
31 $$1 + 7.06e3iT - 2.86e7T^{2}$$
37 $$1 + 1.95e3T + 6.93e7T^{2}$$
41 $$1 - 1.80e3T + 1.15e8T^{2}$$
43 $$1 - 7.88e3T + 1.47e8T^{2}$$
47 $$1 + 6.09e3T + 2.29e8T^{2}$$
53 $$1 - 1.41e4iT - 4.18e8T^{2}$$
59 $$1 - 1.69e4T + 7.14e8T^{2}$$
61 $$1 - 2.97e4iT - 8.44e8T^{2}$$
67 $$1 - 1.36e4T + 1.35e9T^{2}$$
71 $$1 + 3.13e4iT - 1.80e9T^{2}$$
73 $$1 - 9.82e3iT - 2.07e9T^{2}$$
79 $$1 - 9.97e4T + 3.07e9T^{2}$$
83 $$1 + 7.95e4T + 3.93e9T^{2}$$
89 $$1 - 953.T + 5.58e9T^{2}$$
97 $$1 + 1.15e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$