# Properties

 Label 2-147-147.101-c3-0-10 Degree $2$ Conductor $147$ Sign $0.926 - 0.376i$ 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 + (−1.96 − 0.770i)2-s + (1.25 − 5.04i)3-s + (−2.60 − 2.41i)4-s + (10.6 + 7.23i)5-s + (−6.34 + 8.93i)6-s + (3.24 + 18.2i)7-s + (10.5 + 21.9i)8-s + (−23.8 − 12.6i)9-s + (−15.2 − 22.3i)10-s + (7.14 + 47.4i)11-s + (−15.4 + 10.1i)12-s + (−39.7 + 31.7i)13-s + (7.68 − 38.2i)14-s + (49.7 − 44.4i)15-s + (−1.71 − 22.8i)16-s + (74.8 + 23.0i)17-s + ⋯
 L(s)  = 1 + (−0.693 − 0.272i)2-s + (0.241 − 0.970i)3-s + (−0.325 − 0.302i)4-s + (0.948 + 0.646i)5-s + (−0.431 + 0.607i)6-s + (0.175 + 0.984i)7-s + (0.467 + 0.970i)8-s + (−0.883 − 0.468i)9-s + (−0.482 − 0.707i)10-s + (0.195 + 1.29i)11-s + (−0.371 + 0.243i)12-s + (−0.848 + 0.676i)13-s + (0.146 − 0.730i)14-s + (0.856 − 0.764i)15-s + (−0.0267 − 0.357i)16-s + (1.06 + 0.329i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.926 - 0.376i$ Analytic conductor: $$8.67328$$ Root analytic conductor: $$2.94504$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{147} (101, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :3/2),\ 0.926 - 0.376i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.12320 + 0.219648i$$ $$L(\frac12)$$ $$\approx$$ $$1.12320 + 0.219648i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.25 + 5.04i)T$$
7 $$1 + (-3.24 - 18.2i)T$$
good2 $$1 + (1.96 + 0.770i)T + (5.86 + 5.44i)T^{2}$$
5 $$1 + (-10.6 - 7.23i)T + (45.6 + 116. i)T^{2}$$
11 $$1 + (-7.14 - 47.4i)T + (-1.27e3 + 392. i)T^{2}$$
13 $$1 + (39.7 - 31.7i)T + (488. - 2.14e3i)T^{2}$$
17 $$1 + (-74.8 - 23.0i)T + (4.05e3 + 2.76e3i)T^{2}$$
19 $$1 + (-3.72 + 2.15i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (3.58 + 11.6i)T + (-1.00e4 + 6.85e3i)T^{2}$$
29 $$1 + (59.0 - 13.4i)T + (2.19e4 - 1.05e4i)T^{2}$$
31 $$1 + (-243. - 140. i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (165. - 153. i)T + (3.78e3 - 5.05e4i)T^{2}$$
41 $$1 + (-306. + 147. i)T + (4.29e4 - 5.38e4i)T^{2}$$
43 $$1 + (171. + 82.6i)T + (4.95e4 + 6.21e4i)T^{2}$$
47 $$1 + (-128. + 327. i)T + (-7.61e4 - 7.06e4i)T^{2}$$
53 $$1 + (101. - 109. i)T + (-1.11e4 - 1.48e5i)T^{2}$$
59 $$1 + (327. - 223. i)T + (7.50e4 - 1.91e5i)T^{2}$$
61 $$1 + (-114. - 123. i)T + (-1.69e4 + 2.26e5i)T^{2}$$
67 $$1 + (28.2 - 48.8i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + (-493. - 112. i)T + (3.22e5 + 1.55e5i)T^{2}$$
73 $$1 + (-337. + 132. i)T + (2.85e5 - 2.64e5i)T^{2}$$
79 $$1 + (638. + 1.10e3i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + (395. - 495. i)T + (-1.27e5 - 5.57e5i)T^{2}$$
89 $$1 + (-113. - 17.0i)T + (6.73e5 + 2.07e5i)T^{2}$$
97 $$1 - 1.36e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$