# Properties

 Label 2-147-7.2-c5-0-10 Degree $2$ Conductor $147$ Sign $0.991 + 0.126i$ 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 + (4.69 − 8.13i)2-s + (−4.5 − 7.79i)3-s + (−28.0 − 48.6i)4-s + (−35.8 + 62.1i)5-s − 84.5·6-s − 226.·8-s + (−40.5 + 70.1i)9-s + (336. + 583. i)10-s + (280. + 485. i)11-s + (−252. + 437. i)12-s − 533.·13-s + 645.·15-s + (−166. + 288. i)16-s + (502. + 870. i)17-s + (380. + 658. i)18-s + (684. − 1.18e3i)19-s + ⋯
 L(s)  = 1 + (0.829 − 1.43i)2-s + (−0.288 − 0.499i)3-s + (−0.877 − 1.52i)4-s + (−0.641 + 1.11i)5-s − 0.958·6-s − 1.25·8-s + (−0.166 + 0.288i)9-s + (1.06 + 1.84i)10-s + (0.698 + 1.20i)11-s + (−0.506 + 0.877i)12-s − 0.875·13-s + 0.740·15-s + (−0.162 + 0.282i)16-s + (0.422 + 0.730i)17-s + (0.276 + 0.479i)18-s + (0.434 − 0.753i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.991 + 0.126i$ Analytic conductor: $$23.5764$$ Root analytic conductor: $$4.85555$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{147} (79, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :5/2),\ 0.991 + 0.126i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.619999331$$ $$L(\frac12)$$ $$\approx$$ $$1.619999331$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (4.5 + 7.79i)T$$
7 $$1$$
good2 $$1 + (-4.69 + 8.13i)T + (-16 - 27.7i)T^{2}$$
5 $$1 + (35.8 - 62.1i)T + (-1.56e3 - 2.70e3i)T^{2}$$
11 $$1 + (-280. - 485. i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 + 533.T + 3.71e5T^{2}$$
17 $$1 + (-502. - 870. i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (-684. + 1.18e3i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (1.61e3 - 2.79e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + 753.T + 2.05e7T^{2}$$
31 $$1 + (-4.10e3 - 7.10e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (-1.40e3 + 2.43e3i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 + 245.T + 1.15e8T^{2}$$
43 $$1 + 1.75e4T + 1.47e8T^{2}$$
47 $$1 + (8.17e3 - 1.41e4i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (-1.48e4 - 2.56e4i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 + (5.17e3 + 8.96e3i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (-477. + 826. i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (-9.90e3 - 1.71e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 - 6.21e4T + 1.80e9T^{2}$$
73 $$1 + (-1.35e4 - 2.34e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (2.23e4 - 3.87e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + 1.56e4T + 3.93e9T^{2}$$
89 $$1 + (-6.81e3 + 1.18e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 - 1.29e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$