# Properties

 Label 2-3072-48.29-c0-0-5 Degree $2$ Conductor $3072$ Sign $0.923 - 0.382i$ Analytic cond. $1.53312$ Root an. cond. $1.23819$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 − 0.707i)3-s + (1 + i)5-s + 1.41i·7-s − 1.00i·9-s + 1.41·15-s + (1.00 + 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s + (1 − i)29-s − 1.41·31-s + (−1.41 + 1.41i)35-s + (1.00 − 1.00i)45-s − 1.00·49-s + (1 + i)53-s + (1.41 + 1.41i)59-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)3-s + (1 + i)5-s + 1.41i·7-s − 1.00i·9-s + 1.41·15-s + (1.00 + 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s + (1 − i)29-s − 1.41·31-s + (−1.41 + 1.41i)35-s + (1.00 − 1.00i)45-s − 1.00·49-s + (1 + i)53-s + (1.41 + 1.41i)59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3072$$    =    $$2^{10} \cdot 3$$ Sign: $0.923 - 0.382i$ Analytic conductor: $$1.53312$$ Root analytic conductor: $$1.23819$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3072} (257, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3072,\ (\ :0),\ 0.923 - 0.382i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.887843188$$ $$L(\frac12)$$ $$\approx$$ $$1.887843188$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-0.707 + 0.707i)T$$
good5 $$1 + (-1 - i)T + iT^{2}$$
7 $$1 - 1.41iT - T^{2}$$
11 $$1 + 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 + 1.41T + T^{2}$$
37 $$1 - iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-1 - i)T + iT^{2}$$
59 $$1 + (-1.41 - 1.41i)T + iT^{2}$$
61 $$1 + iT^{2}$$
67 $$1 + iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 + 1.41T + T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 + 2T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$