# Properties

 Label 2-147-7.2-c3-0-8 Degree $2$ Conductor $147$ Sign $0.0725 - 0.997i$ 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 + (0.207 − 0.358i)2-s + (1.5 + 2.59i)3-s + (3.91 + 6.77i)4-s + (−0.0502 + 0.0870i)5-s + 1.24·6-s + 6.55·8-s + (−4.5 + 7.79i)9-s + (0.0208 + 0.0360i)10-s + (21.9 + 38.0i)11-s + (−11.7 + 20.3i)12-s + 16.6·13-s − 0.301·15-s + (−29.9 + 51.8i)16-s + (−60.8 − 105. i)17-s + (1.86 + 3.22i)18-s + (−63.5 + 110. i)19-s + ⋯
 L(s)  = 1 + (0.0732 − 0.126i)2-s + (0.288 + 0.499i)3-s + (0.489 + 0.847i)4-s + (−0.00449 + 0.00778i)5-s + 0.0845·6-s + 0.289·8-s + (−0.166 + 0.288i)9-s + (0.000658 + 0.00114i)10-s + (0.602 + 1.04i)11-s + (−0.282 + 0.489i)12-s + 0.355·13-s − 0.00519·15-s + (−0.468 + 0.810i)16-s + (−0.867 − 1.50i)17-s + (0.0244 + 0.0422i)18-s + (−0.767 + 1.32i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.0725 - 0.997i$ Analytic conductor: $$8.67328$$ Root analytic conductor: $$2.94504$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{147} (79, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :3/2),\ 0.0725 - 0.997i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.49263 + 1.38798i$$ $$L(\frac12)$$ $$\approx$$ $$1.49263 + 1.38798i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.5 - 2.59i)T$$
7 $$1$$
good2 $$1 + (-0.207 + 0.358i)T + (-4 - 6.92i)T^{2}$$
5 $$1 + (0.0502 - 0.0870i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-21.9 - 38.0i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 - 16.6T + 2.19e3T^{2}$$
17 $$1 + (60.8 + 105. i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (63.5 - 110. i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (26.7 - 46.4i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 235.T + 2.43e4T^{2}$$
31 $$1 + (9.35 + 16.2i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-95.9 + 166. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 319.T + 6.89e4T^{2}$$
43 $$1 + 218.T + 7.95e4T^{2}$$
47 $$1 + (-200. + 347. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (321. + 556. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (5.80 + 10.0i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (6.12 - 10.6i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (334. + 579. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 822.T + 3.57e5T^{2}$$
73 $$1 + (-257. - 446. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-402. + 697. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 - 394.T + 5.71e5T^{2}$$
89 $$1 + (-336. + 583. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 - 1.09e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$