Properties

 Label 2-448-112.69-c0-0-0 Degree $2$ Conductor $448$ Sign $0.923 + 0.382i$ Analytic cond. $0.223581$ Root an. cond. $0.472843$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − i·7-s − i·9-s + (1 + i)11-s + i·25-s + (−1 + i)29-s + (−1 − i)37-s + (−1 − i)43-s − 49-s + (1 + i)53-s − 63-s + (−1 + i)67-s + 2i·71-s + (1 − i)77-s − 81-s + (1 − i)99-s + ⋯
 L(s)  = 1 − i·7-s − i·9-s + (1 + i)11-s + i·25-s + (−1 + i)29-s + (−1 − i)37-s + (−1 − i)43-s − 49-s + (1 + i)53-s − 63-s + (−1 + i)67-s + 2i·71-s + (1 − i)77-s − 81-s + (1 − i)99-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$448$$    =    $$2^{6} \cdot 7$$ Sign: $0.923 + 0.382i$ Analytic conductor: $$0.223581$$ Root analytic conductor: $$0.472843$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{448} (433, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 448,\ (\ :0),\ 0.923 + 0.382i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8788556582$$ $$L(\frac12)$$ $$\approx$$ $$0.8788556582$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + iT$$
good3 $$1 + iT^{2}$$
5 $$1 - iT^{2}$$
11 $$1 + (-1 - i)T + iT^{2}$$
13 $$1 + iT^{2}$$
17 $$1 - T^{2}$$
19 $$1 + iT^{2}$$
23 $$1 - T^{2}$$
29 $$1 + (1 - i)T - iT^{2}$$
31 $$1 - T^{2}$$
37 $$1 + (1 + i)T + iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 + (1 + i)T + iT^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-1 - i)T + iT^{2}$$
59 $$1 - iT^{2}$$
61 $$1 + iT^{2}$$
67 $$1 + (1 - i)T - iT^{2}$$
71 $$1 - 2iT - T^{2}$$
73 $$1 + T^{2}$$
79 $$1 + T^{2}$$
83 $$1 + iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$