# Properties

 Label 2-147-21.20-c5-0-25 Degree $2$ Conductor $147$ Sign $0.973 + 0.226i$ 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 + 2.13i·2-s + (−9.16 − 12.6i)3-s + 27.4·4-s − 95.0·5-s + (26.8 − 19.5i)6-s + 126. i·8-s + (−75.1 + 231. i)9-s − 202. i·10-s + 130. i·11-s + (−251. − 346. i)12-s + 14.6i·13-s + (870. + 1.19e3i)15-s + 608.·16-s + 640.·17-s + (−492. − 160. i)18-s + 542. i·19-s + ⋯
 L(s)  = 1 + 0.377i·2-s + (−0.587 − 0.809i)3-s + 0.857·4-s − 1.70·5-s + (0.305 − 0.221i)6-s + 0.700i·8-s + (−0.309 + 0.951i)9-s − 0.641i·10-s + 0.325i·11-s + (−0.504 − 0.694i)12-s + 0.0240i·13-s + (0.999 + 1.37i)15-s + 0.593·16-s + 0.537·17-s + (−0.358 − 0.116i)18-s + 0.344i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.973 + 0.226i$ 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: $$(2,\ 147,\ (\ :5/2),\ 0.973 + 0.226i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.274191190$$ $$L(\frac12)$$ $$\approx$$ $$1.274191190$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (9.16 + 12.6i)T$$
7 $$1$$
good2 $$1 - 2.13iT - 32T^{2}$$
5 $$1 + 95.0T + 3.12e3T^{2}$$
11 $$1 - 130. iT - 1.61e5T^{2}$$
13 $$1 - 14.6iT - 3.71e5T^{2}$$
17 $$1 - 640.T + 1.41e6T^{2}$$
19 $$1 - 542. iT - 2.47e6T^{2}$$
23 $$1 + 4.09e3iT - 6.43e6T^{2}$$
29 $$1 + 3.59e3iT - 2.05e7T^{2}$$
31 $$1 + 2.90e3iT - 2.86e7T^{2}$$
37 $$1 + 4.34e3T + 6.93e7T^{2}$$
41 $$1 - 1.09e4T + 1.15e8T^{2}$$
43 $$1 - 2.05e4T + 1.47e8T^{2}$$
47 $$1 - 8.81e3T + 2.29e8T^{2}$$
53 $$1 - 6.97e3iT - 4.18e8T^{2}$$
59 $$1 - 4.00e4T + 7.14e8T^{2}$$
61 $$1 - 3.32e4iT - 8.44e8T^{2}$$
67 $$1 - 6.43e3T + 1.35e9T^{2}$$
71 $$1 - 3.24e4iT - 1.80e9T^{2}$$
73 $$1 + 1.27e4iT - 2.07e9T^{2}$$
79 $$1 - 8.59e3T + 3.07e9T^{2}$$
83 $$1 - 3.16e4T + 3.93e9T^{2}$$
89 $$1 + 3.19e4T + 5.58e9T^{2}$$
97 $$1 + 1.43e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$