# Properties

 Label 2-3072-96.53-c0-0-1 Degree $2$ Conductor $3072$ Sign $0.980 - 0.195i$ 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 + (−0.923 + 0.382i)3-s + (−0.541 − 0.541i)7-s + (0.707 − 0.707i)9-s + (0.707 + 1.70i)13-s + (−0.541 − 1.30i)19-s + (0.707 + 0.292i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + 1.84·31-s + (−0.292 + 0.707i)37-s + (−1.30 − 1.30i)39-s + (1.30 + 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (1.70 − 0.707i)61-s + ⋯
 L(s)  = 1 + (−0.923 + 0.382i)3-s + (−0.541 − 0.541i)7-s + (0.707 − 0.707i)9-s + (0.707 + 1.70i)13-s + (−0.541 − 1.30i)19-s + (0.707 + 0.292i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + 1.84·31-s + (−0.292 + 0.707i)37-s + (−1.30 − 1.30i)39-s + (1.30 + 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (1.70 − 0.707i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3072$$    =    $$2^{10} \cdot 3$$ Sign: $0.980 - 0.195i$ Analytic conductor: $$1.53312$$ Root analytic conductor: $$1.23819$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3072} (1409, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3072,\ (\ :0),\ 0.980 - 0.195i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8536389875$$ $$L(\frac12)$$ $$\approx$$ $$0.8536389875$$ $$L(1)$$ 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 + (0.923 - 0.382i)T$$
good5 $$1 + (0.707 + 0.707i)T^{2}$$
7 $$1 + (0.541 + 0.541i)T + iT^{2}$$
11 $$1 + (-0.707 - 0.707i)T^{2}$$
13 $$1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2}$$
17 $$1 + T^{2}$$
19 $$1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2}$$
23 $$1 + iT^{2}$$
29 $$1 + (-0.707 + 0.707i)T^{2}$$
31 $$1 - 1.84T + T^{2}$$
37 $$1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2}$$
41 $$1 + iT^{2}$$
43 $$1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2}$$
47 $$1 + T^{2}$$
53 $$1 + (-0.707 - 0.707i)T^{2}$$
59 $$1 + (0.707 + 0.707i)T^{2}$$
61 $$1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2}$$
67 $$1 + (0.707 - 0.707i)T^{2}$$
71 $$1 - iT^{2}$$
73 $$1 + (-1 + i)T - iT^{2}$$
79 $$1 - 1.84iT - T^{2}$$
83 $$1 + (0.707 - 0.707i)T^{2}$$
89 $$1 - iT^{2}$$
97 $$1 - 1.41T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$