# Properties

 Degree $2$ Conductor $168$ Sign $0.605 + 0.795i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)3-s + (−1 − 1.73i)5-s + (2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (3 − 5.19i)11-s − 3·13-s − 1.99·15-s + (−2 + 3.46i)17-s + (2.5 + 4.33i)19-s + (2 − 1.73i)21-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s − 0.999·27-s − 4·29-s + (−3.5 + 6.06i)31-s + ⋯
 L(s)  = 1 + (0.288 − 0.499i)3-s + (−0.447 − 0.774i)5-s + (0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.904 − 1.56i)11-s − 0.832·13-s − 0.516·15-s + (−0.485 + 0.840i)17-s + (0.573 + 0.993i)19-s + (0.436 − 0.377i)21-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s − 0.192·27-s − 0.742·29-s + (−0.628 + 1.08i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$168$$    =    $$2^{3} \cdot 3 \cdot 7$$ Sign: $0.605 + 0.795i$ Motivic weight: $$1$$ Character: $\chi_{168} (25, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 168,\ (\ :1/2),\ 0.605 + 0.795i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.11967 - 0.555054i$$ $$L(\frac12)$$ $$\approx$$ $$1.11967 - 0.555054i$$ $$L(\frac{3}{2})$$ 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 + (-0.5 + 0.866i)T$$
7 $$1 + (-2.5 - 0.866i)T$$
good5 $$1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 3T + 13T^{2}$$
17 $$1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 4T + 29T^{2}$$
31 $$1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 + T + 43T^{2}$$
47 $$1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-7.5 + 12.9i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 6T + 71T^{2}$$
73 $$1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 6T + 83T^{2}$$
89 $$1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 14T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$