# Properties

 Degree $2$ Conductor $7056$ Sign $-0.944 + 0.327i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.87i·5-s − 3.87i·11-s − 3.46i·13-s + 4·19-s − 7.74i·23-s − 10.0·25-s + 6.70·29-s − 31-s + 4·37-s − 7.74i·41-s − 6.92i·43-s + 13.4·47-s − 6.70·53-s − 15.0·55-s + 6.70·59-s + ⋯
 L(s)  = 1 − 1.73i·5-s − 1.16i·11-s − 0.960i·13-s + 0.917·19-s − 1.61i·23-s − 2.00·25-s + 1.24·29-s − 0.179·31-s + 0.657·37-s − 1.20i·41-s − 1.05i·43-s + 1.95·47-s − 0.921·53-s − 2.02·55-s + 0.873·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$7056$$    =    $$2^{4} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-0.944 + 0.327i$ Motivic weight: $$1$$ Character: $\chi_{7056} (1567, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 7056,\ (\ :1/2),\ -0.944 + 0.327i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.015908699$$ $$L(\frac12)$$ $$\approx$$ $$2.015908699$$ $$L(\frac{3}{2})$$ 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$$
7 $$1$$
good5 $$1 + 3.87iT - 5T^{2}$$
11 $$1 + 3.87iT - 11T^{2}$$
13 $$1 + 3.46iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 + 7.74iT - 23T^{2}$$
29 $$1 - 6.70T + 29T^{2}$$
31 $$1 + T + 31T^{2}$$
37 $$1 - 4T + 37T^{2}$$
41 $$1 + 7.74iT - 41T^{2}$$
43 $$1 + 6.92iT - 43T^{2}$$
47 $$1 - 13.4T + 47T^{2}$$
53 $$1 + 6.70T + 53T^{2}$$
59 $$1 - 6.70T + 59T^{2}$$
61 $$1 - 10.3iT - 61T^{2}$$
67 $$1 + 6.92iT - 67T^{2}$$
71 $$1 - 7.74iT - 71T^{2}$$
73 $$1 - 6.92iT - 73T^{2}$$
79 $$1 + 12.1iT - 79T^{2}$$
83 $$1 - 6.70T + 83T^{2}$$
89 $$1 + 7.74iT - 89T^{2}$$
97 $$1 - 5.19iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$