# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.74·5-s − 5.29·11-s − 4.24·13-s − 3.74·17-s − 2.82·19-s + 5.29·23-s + 9·25-s − 5.29·29-s − 8.48·31-s + 4·37-s − 3.74·41-s − 8·43-s − 7.48·47-s − 10.5·53-s + 19.7·55-s − 7.48·59-s + 9.89·61-s + 15.8·65-s − 12·67-s − 15.8·71-s + 1.41·73-s + 4·79-s − 14.9·83-s + 14·85-s + 3.74·89-s + 10.5·95-s − 9.89·97-s + ⋯
 L(s)  = 1 − 1.67·5-s − 1.59·11-s − 1.17·13-s − 0.907·17-s − 0.648·19-s + 1.10·23-s + 1.80·25-s − 0.982·29-s − 1.52·31-s + 0.657·37-s − 0.584·41-s − 1.21·43-s − 1.09·47-s − 1.45·53-s + 2.66·55-s − 0.974·59-s + 1.26·61-s + 1.96·65-s − 1.46·67-s − 1.88·71-s + 0.165·73-s + 0.450·79-s − 1.64·83-s + 1.51·85-s + 0.396·89-s + 1.08·95-s − 1.00·97-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 = $$0$$ Selberg data = $$(2,\ 7056,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.04431082381$$ $$L(\frac12)$$ $$\approx$$ $$0.04431082381$$ $$L(\frac{3}{2})$$ 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 + 3.74T + 5T^{2}$$
11 $$1 + 5.29T + 11T^{2}$$
13 $$1 + 4.24T + 13T^{2}$$
17 $$1 + 3.74T + 17T^{2}$$
19 $$1 + 2.82T + 19T^{2}$$
23 $$1 - 5.29T + 23T^{2}$$
29 $$1 + 5.29T + 29T^{2}$$
31 $$1 + 8.48T + 31T^{2}$$
37 $$1 - 4T + 37T^{2}$$
41 $$1 + 3.74T + 41T^{2}$$
43 $$1 + 8T + 43T^{2}$$
47 $$1 + 7.48T + 47T^{2}$$
53 $$1 + 10.5T + 53T^{2}$$
59 $$1 + 7.48T + 59T^{2}$$
61 $$1 - 9.89T + 61T^{2}$$
67 $$1 + 12T + 67T^{2}$$
71 $$1 + 15.8T + 71T^{2}$$
73 $$1 - 1.41T + 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 + 14.9T + 83T^{2}$$
89 $$1 - 3.74T + 89T^{2}$$
97 $$1 + 9.89T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}