# Properties

 Label 2-168-7.4-c3-0-9 Degree $2$ Conductor $168$ Sign $-0.514 + 0.857i$ Analytic cond. $9.91232$ Root an. cond. $3.14838$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.5 + 2.59i)3-s + (−0.0642 − 0.111i)5-s + (−0.866 − 18.4i)7-s + (−4.5 − 7.79i)9-s + (−27.0 + 46.7i)11-s − 50.2·13-s + 0.385·15-s + (65.7 − 113. i)17-s + (−45.7 − 79.2i)19-s + (49.3 + 25.4i)21-s + (−89.7 − 155. i)23-s + (62.4 − 108. i)25-s + 27·27-s − 69.8·29-s + (−163. + 283. i)31-s + ⋯
 L(s)  = 1 + (−0.288 + 0.499i)3-s + (−0.00575 − 0.00996i)5-s + (−0.0467 − 0.998i)7-s + (−0.166 − 0.288i)9-s + (−0.740 + 1.28i)11-s − 1.07·13-s + 0.00664·15-s + (0.937 − 1.62i)17-s + (−0.552 − 0.956i)19-s + (0.512 + 0.264i)21-s + (−0.813 − 1.40i)23-s + (0.499 − 0.865i)25-s + 0.192·27-s − 0.447·29-s + (−0.946 + 1.63i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$168$$    =    $$2^{3} \cdot 3 \cdot 7$$ Sign: $-0.514 + 0.857i$ Analytic conductor: $$9.91232$$ Root analytic conductor: $$3.14838$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{168} (25, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 168,\ (\ :3/2),\ -0.514 + 0.857i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.292775 - 0.516954i$$ $$L(\frac12)$$ $$\approx$$ $$0.292775 - 0.516954i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (1.5 - 2.59i)T$$
7 $$1 + (0.866 + 18.4i)T$$
good5 $$1 + (0.0642 + 0.111i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (27.0 - 46.7i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + 50.2T + 2.19e3T^{2}$$
17 $$1 + (-65.7 + 113. i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (45.7 + 79.2i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (89.7 + 155. i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 69.8T + 2.43e4T^{2}$$
31 $$1 + (163. - 283. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (150. + 261. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 - 296.T + 6.89e4T^{2}$$
43 $$1 + 144.T + 7.95e4T^{2}$$
47 $$1 + (-180. - 311. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (0.917 - 1.58i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (-26.6 + 46.1i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-54.0 - 93.6i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (421. - 729. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 241.T + 3.57e5T^{2}$$
73 $$1 + (-103. + 179. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (279. + 484. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 986.T + 5.71e5T^{2}$$
89 $$1 + (-221. - 383. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + 740.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$