# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 7^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.27·5-s + 3.27·11-s − 6.27·13-s − 4·17-s − 6.27·19-s + 4·23-s + 5.72·25-s − 5.27·29-s − 31-s − 2.27·37-s − 4.54·41-s − 0.274·43-s + 6·47-s − 9.27·53-s + 10.7·55-s + 1.27·59-s − 10·61-s − 20.5·65-s + 0.274·67-s + 2·71-s + 4.27·73-s − 11.5·79-s − 7.27·83-s − 13.0·85-s − 10.5·89-s − 20.5·95-s − 8.72·97-s + ⋯
 L(s)  = 1 + 1.46·5-s + 0.987·11-s − 1.74·13-s − 0.970·17-s − 1.43·19-s + 0.834·23-s + 1.14·25-s − 0.979·29-s − 0.179·31-s − 0.373·37-s − 0.710·41-s − 0.0419·43-s + 0.875·47-s − 1.27·53-s + 1.44·55-s + 0.165·59-s − 1.28·61-s − 2.54·65-s + 0.0335·67-s + 0.237·71-s + 0.500·73-s − 1.29·79-s − 0.798·83-s − 1.42·85-s − 1.11·89-s − 2.10·95-s − 0.885·97-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

 $$d$$ = $$2$$ $$N$$ = $$7056$$    =    $$2^{4} \cdot 3^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{7056} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 7056,\ (\ :1/2),\ -1)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1$$
good5 $$1 - 3.27T + 5T^{2}$$
11 $$1 - 3.27T + 11T^{2}$$
13 $$1 + 6.27T + 13T^{2}$$
17 $$1 + 4T + 17T^{2}$$
19 $$1 + 6.27T + 19T^{2}$$
23 $$1 - 4T + 23T^{2}$$
29 $$1 + 5.27T + 29T^{2}$$
31 $$1 + T + 31T^{2}$$
37 $$1 + 2.27T + 37T^{2}$$
41 $$1 + 4.54T + 41T^{2}$$
43 $$1 + 0.274T + 43T^{2}$$
47 $$1 - 6T + 47T^{2}$$
53 $$1 + 9.27T + 53T^{2}$$
59 $$1 - 1.27T + 59T^{2}$$
61 $$1 + 10T + 61T^{2}$$
67 $$1 - 0.274T + 67T^{2}$$
71 $$1 - 2T + 71T^{2}$$
73 $$1 - 4.27T + 73T^{2}$$
79 $$1 + 11.5T + 79T^{2}$$
83 $$1 + 7.27T + 83T^{2}$$
89 $$1 + 10.5T + 89T^{2}$$
97 $$1 + 8.72T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}