# Properties

 Label 2-84e2-1.1-c1-0-68 Degree $2$ Conductor $7056$ Sign $-1$ Analytic cond. $56.3424$ Root an. cond. $7.50616$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.29·11-s + 5.29·23-s − 5·25-s + 10.5·29-s + 6·37-s − 12·43-s + 10.5·53-s − 4·67-s − 5.29·71-s − 8·79-s − 5.29·107-s − 18·109-s − 21.1·113-s + ⋯
 L(s)  = 1 − 1.59·11-s + 1.10·23-s − 25-s + 1.96·29-s + 0.986·37-s − 1.82·43-s + 1.45·53-s − 0.488·67-s − 0.627·71-s − 0.900·79-s − 0.511·107-s − 1.72·109-s − 1.99·113-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: $-1$ Analytic conductor: $$56.3424$$ Root analytic conductor: $$7.50616$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial 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 + 5T^{2}$$
11 $$1 + 5.29T + 11T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 - 5.29T + 23T^{2}$$
29 $$1 - 10.5T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 6T + 37T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 + 12T + 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 - 10.5T + 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + 61T^{2}$$
67 $$1 + 4T + 67T^{2}$$
71 $$1 + 5.29T + 71T^{2}$$
73 $$1 + 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + 89T^{2}$$
97 $$1 + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$