# Properties

 Degree $2$ Conductor $7056$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.585·5-s − 2·11-s + 5.41·13-s − 6.24·17-s − 2.82·19-s + 3.65·23-s − 4.65·25-s + 1.17·29-s + 6.82·31-s − 4·37-s + 2.24·41-s + 5.65·43-s + 2.82·47-s + 2·53-s + 1.17·55-s − 6.82·59-s + 3.75·61-s − 3.17·65-s − 5.65·67-s − 13.3·71-s − 5.89·73-s − 2.34·79-s − 15.3·83-s + 3.65·85-s + 5.75·89-s + 1.65·95-s + 5.41·97-s + ⋯
 L(s)  = 1 − 0.261·5-s − 0.603·11-s + 1.50·13-s − 1.51·17-s − 0.648·19-s + 0.762·23-s − 0.931·25-s + 0.217·29-s + 1.22·31-s − 0.657·37-s + 0.350·41-s + 0.862·43-s + 0.412·47-s + 0.274·53-s + 0.157·55-s − 0.888·59-s + 0.481·61-s − 0.393·65-s − 0.691·67-s − 1.58·71-s − 0.690·73-s − 0.263·79-s − 1.68·83-s + 0.396·85-s + 0.610·89-s + 0.169·95-s + 0.549·97-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$7056$$    =    $$2^{4} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{7056} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 7056,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 + 0.585T + 5T^{2}$$
11 $$1 + 2T + 11T^{2}$$
13 $$1 - 5.41T + 13T^{2}$$
17 $$1 + 6.24T + 17T^{2}$$
19 $$1 + 2.82T + 19T^{2}$$
23 $$1 - 3.65T + 23T^{2}$$
29 $$1 - 1.17T + 29T^{2}$$
31 $$1 - 6.82T + 31T^{2}$$
37 $$1 + 4T + 37T^{2}$$
41 $$1 - 2.24T + 41T^{2}$$
43 $$1 - 5.65T + 43T^{2}$$
47 $$1 - 2.82T + 47T^{2}$$
53 $$1 - 2T + 53T^{2}$$
59 $$1 + 6.82T + 59T^{2}$$
61 $$1 - 3.75T + 61T^{2}$$
67 $$1 + 5.65T + 67T^{2}$$
71 $$1 + 13.3T + 71T^{2}$$
73 $$1 + 5.89T + 73T^{2}$$
79 $$1 + 2.34T + 79T^{2}$$
83 $$1 + 15.3T + 83T^{2}$$
89 $$1 - 5.75T + 89T^{2}$$
97 $$1 - 5.41T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$