# Properties

 Label 2-448-7.3-c2-0-22 Degree $2$ Conductor $448$ Sign $0.0725 + 0.997i$ Analytic cond. $12.2071$ Root an. cond. $3.49386$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.621 − 0.358i)3-s + (5.74 − 3.31i)5-s + (6.24 − 3.16i)7-s + (−4.24 − 7.34i)9-s + (−2.37 + 4.11i)11-s − 15.2i·13-s − 4.75·15-s + (−3.25 − 1.88i)17-s + (−3.62 + 2.09i)19-s + (−5.01 − 0.271i)21-s + (13.8 + 24.0i)23-s + (9.48 − 16.4i)25-s + 12.5i·27-s − 3.51·29-s + (−42.3 − 24.4i)31-s + ⋯
 L(s)  = 1 + (−0.207 − 0.119i)3-s + (1.14 − 0.663i)5-s + (0.891 − 0.452i)7-s + (−0.471 − 0.816i)9-s + (−0.216 + 0.374i)11-s − 1.17i·13-s − 0.317·15-s + (−0.191 − 0.110i)17-s + (−0.190 + 0.110i)19-s + (−0.238 − 0.0129i)21-s + (0.602 + 1.04i)23-s + (0.379 − 0.657i)25-s + 0.464i·27-s − 0.121·29-s + (−1.36 − 0.788i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$448$$    =    $$2^{6} \cdot 7$$ Sign: $0.0725 + 0.997i$ Analytic conductor: $$12.2071$$ Root analytic conductor: $$3.49386$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{448} (129, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 448,\ (\ :1),\ 0.0725 + 0.997i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.917692017$$ $$L(\frac12)$$ $$\approx$$ $$1.917692017$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + (-6.24 + 3.16i)T$$
good3 $$1 + (0.621 + 0.358i)T + (4.5 + 7.79i)T^{2}$$
5 $$1 + (-5.74 + 3.31i)T + (12.5 - 21.6i)T^{2}$$
11 $$1 + (2.37 - 4.11i)T + (-60.5 - 104. i)T^{2}$$
13 $$1 + 15.2iT - 169T^{2}$$
17 $$1 + (3.25 + 1.88i)T + (144.5 + 250. i)T^{2}$$
19 $$1 + (3.62 - 2.09i)T + (180.5 - 312. i)T^{2}$$
23 $$1 + (-13.8 - 24.0i)T + (-264.5 + 458. i)T^{2}$$
29 $$1 + 3.51T + 841T^{2}$$
31 $$1 + (42.3 + 24.4i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (1.47 + 2.54i)T + (-684.5 + 1.18e3i)T^{2}$$
41 $$1 + 27.9iT - 1.68e3T^{2}$$
43 $$1 - 10.4T + 1.84e3T^{2}$$
47 $$1 + (-45.6 + 26.3i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (-27.9 + 48.4i)T + (-1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (33.5 + 19.3i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (-78.3 + 45.2i)T + (1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (17.3 - 29.9i)T + (-2.24e3 - 3.88e3i)T^{2}$$
71 $$1 - 36.4T + 5.04e3T^{2}$$
73 $$1 + (-45.5 - 26.3i)T + (2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (-16.8 - 29.2i)T + (-3.12e3 + 5.40e3i)T^{2}$$
83 $$1 - 127. iT - 6.88e3T^{2}$$
89 $$1 + (43.5 - 25.1i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 - 101. iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$