# Properties

 Degree $2$ Conductor $672$ Sign $-0.985 + 0.169i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.41 − i)3-s + 4.24i·5-s + i·7-s + (1.00 + 2.82i)9-s − 4.24·11-s − 2·13-s + (4.24 − 6i)15-s − 7.07i·17-s − 4i·19-s + (1 − 1.41i)21-s − 1.41·23-s − 12.9·25-s + (1.41 − 5.00i)27-s + 2.82i·29-s − 2i·31-s + ⋯
 L(s)  = 1 + (−0.816 − 0.577i)3-s + 1.89i·5-s + 0.377i·7-s + (0.333 + 0.942i)9-s − 1.27·11-s − 0.554·13-s + (1.09 − 1.54i)15-s − 1.71i·17-s − 0.917i·19-s + (0.218 − 0.308i)21-s − 0.294·23-s − 2.59·25-s + (0.272 − 0.962i)27-s + 0.525i·29-s − 0.359i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$672$$    =    $$2^{5} \cdot 3 \cdot 7$$ Sign: $-0.985 + 0.169i$ Motivic weight: $$1$$ Character: $\chi_{672} (575, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 672,\ (\ :1/2),\ -0.985 + 0.169i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.0147008 - 0.172617i$$ $$L(\frac12)$$ $$\approx$$ $$0.0147008 - 0.172617i$$ $$L(\frac{3}{2})$$ 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 + (1.41 + i)T$$
7 $$1 - iT$$
good5 $$1 - 4.24iT - 5T^{2}$$
11 $$1 + 4.24T + 11T^{2}$$
13 $$1 + 2T + 13T^{2}$$
17 $$1 + 7.07iT - 17T^{2}$$
19 $$1 + 4iT - 19T^{2}$$
23 $$1 + 1.41T + 23T^{2}$$
29 $$1 - 2.82iT - 29T^{2}$$
31 $$1 + 2iT - 31T^{2}$$
37 $$1 + 4T + 37T^{2}$$
41 $$1 - 1.41iT - 41T^{2}$$
43 $$1 - 8iT - 43T^{2}$$
47 $$1 - 2.82T + 47T^{2}$$
53 $$1 + 5.65iT - 53T^{2}$$
59 $$1 + 11.3T + 59T^{2}$$
61 $$1 + 2T + 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 + 12.7T + 71T^{2}$$
73 $$1 - 14T + 73T^{2}$$
79 $$1 - 4iT - 79T^{2}$$
83 $$1 + 2.82T + 83T^{2}$$
89 $$1 - 9.89iT - 89T^{2}$$
97 $$1 + 10T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$