# Properties

 Degree $2$ Conductor $672$ Sign $-0.386 - 0.922i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)3-s + (−2.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + 13-s + (1 + 1.73i)17-s + (−2.5 + 4.33i)19-s + (−2 − 1.73i)21-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s − 0.999·27-s − 8·29-s + (1.5 + 2.59i)31-s + (−0.999 + 1.73i)33-s + (4.5 − 7.79i)37-s + ⋯
 L(s)  = 1 + (0.288 + 0.499i)3-s + (−0.944 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + 0.277·13-s + (0.242 + 0.420i)17-s + (−0.573 + 0.993i)19-s + (−0.436 − 0.377i)21-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s − 0.192·27-s − 1.48·29-s + (0.269 + 0.466i)31-s + (−0.174 + 0.301i)33-s + (0.739 − 1.28i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$672$$    =    $$2^{5} \cdot 3 \cdot 7$$ Sign: $-0.386 - 0.922i$ Motivic weight: $$1$$ Character: $\chi_{672} (289, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 672,\ (\ :1/2),\ -0.386 - 0.922i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.665899 + 1.00107i$$ $$L(\frac12)$$ $$\approx$$ $$0.665899 + 1.00107i$$ $$L(\frac{3}{2})$$ 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 + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 - T + 13T^{2}$$
17 $$1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 8T + 29T^{2}$$
31 $$1 + (-1.5 - 2.59i)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 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 - 4T + 71T^{2}$$
73 $$1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 - 18T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$