# Properties

 Label 2-147-7.4-c5-0-1 Degree $2$ Conductor $147$ Sign $-0.386 + 0.922i$ 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 + (4.22 + 7.31i)2-s + (−4.5 + 7.79i)3-s + (−19.6 + 34.0i)4-s + (18 + 31.1i)5-s − 76.0·6-s − 61.9·8-s + (−40.5 − 70.1i)9-s + (−152. + 263. i)10-s + (−147. + 255. i)11-s + (−177. − 306. i)12-s − 1.14e3·13-s − 324·15-s + (367. + 636. i)16-s + (−516. + 894. i)17-s + (342. − 592. i)18-s + (−1.05e3 − 1.82e3i)19-s + ⋯
 L(s)  = 1 + (0.746 + 1.29i)2-s + (−0.288 + 0.499i)3-s + (−0.614 + 1.06i)4-s + (0.321 + 0.557i)5-s − 0.862·6-s − 0.342·8-s + (−0.166 − 0.288i)9-s + (−0.480 + 0.832i)10-s + (−0.368 + 0.637i)11-s + (−0.354 − 0.614i)12-s − 1.88·13-s − 0.371·15-s + (0.359 + 0.621i)16-s + (−0.433 + 0.750i)17-s + (0.248 − 0.431i)18-s + (−0.669 − 1.16i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $-0.386 + 0.922i$ Analytic conductor: $$23.5764$$ Root analytic conductor: $$4.85555$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{147} (67, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :5/2),\ -0.386 + 0.922i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.520598412$$ $$L(\frac12)$$ $$\approx$$ $$1.520598412$$ $$L(\frac{7}{2})$$ 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 + (-4.22 - 7.31i)T + (-16 + 27.7i)T^{2}$$
5 $$1 + (-18 - 31.1i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (147. - 255. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + 1.14e3T + 3.71e5T^{2}$$
17 $$1 + (516. - 894. i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (1.05e3 + 1.82e3i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (-320. - 555. i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 - 7.63e3T + 2.05e7T^{2}$$
31 $$1 + (-483. + 837. i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (-886. - 1.53e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 + 1.19e4T + 1.15e8T^{2}$$
43 $$1 + 1.98e4T + 1.47e8T^{2}$$
47 $$1 + (-1.39e4 - 2.42e4i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (-3.55e3 + 6.16e3i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-1.04e4 + 1.80e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (-1.19e4 - 2.06e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (1.73e4 - 3.00e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 2.84e4T + 1.80e9T^{2}$$
73 $$1 + (-7.64e3 + 1.32e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (-3.65e4 - 6.32e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 3.03e4T + 3.93e9T^{2}$$
89 $$1 + (-1.80e4 - 3.12e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 + 1.53e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$