# Properties

 Label 2-28e2-1.1-c3-0-47 Degree $2$ Conductor $784$ Sign $-1$ Analytic cond. $46.2574$ Root an. cond. $6.80128$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 7.07·3-s − 19.7·5-s + 23.0·9-s + 14·11-s + 50.9·13-s − 140·15-s + 1.41·17-s + 1.41·19-s − 140·23-s + 267·25-s − 28.2·27-s − 286·29-s + 93.3·31-s + 98.9·33-s − 38·37-s + 360·39-s − 125.·41-s + 34·43-s − 455.·45-s − 523.·47-s + 10.0·51-s − 74·53-s − 277.·55-s + 10.0·57-s − 434.·59-s + 14.1·61-s − 1.00e3·65-s + ⋯
 L(s)  = 1 + 1.36·3-s − 1.77·5-s + 0.851·9-s + 0.383·11-s + 1.08·13-s − 2.40·15-s + 0.0201·17-s + 0.0170·19-s − 1.26·23-s + 2.13·25-s − 0.201·27-s − 1.83·29-s + 0.540·31-s + 0.522·33-s − 0.168·37-s + 1.47·39-s − 0.479·41-s + 0.120·43-s − 1.50·45-s − 1.62·47-s + 0.0274·51-s − 0.191·53-s − 0.679·55-s + 0.0232·57-s − 0.958·59-s + 0.0296·61-s − 1.92·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$784$$    =    $$2^{4} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$46.2574$$ Root analytic conductor: $$6.80128$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 784,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1$$
good3 $$1 - 7.07T + 27T^{2}$$
5 $$1 + 19.7T + 125T^{2}$$
11 $$1 - 14T + 1.33e3T^{2}$$
13 $$1 - 50.9T + 2.19e3T^{2}$$
17 $$1 - 1.41T + 4.91e3T^{2}$$
19 $$1 - 1.41T + 6.85e3T^{2}$$
23 $$1 + 140T + 1.21e4T^{2}$$
29 $$1 + 286T + 2.43e4T^{2}$$
31 $$1 - 93.3T + 2.97e4T^{2}$$
37 $$1 + 38T + 5.06e4T^{2}$$
41 $$1 + 125.T + 6.89e4T^{2}$$
43 $$1 - 34T + 7.95e4T^{2}$$
47 $$1 + 523.T + 1.03e5T^{2}$$
53 $$1 + 74T + 1.48e5T^{2}$$
59 $$1 + 434.T + 2.05e5T^{2}$$
61 $$1 - 14.1T + 2.26e5T^{2}$$
67 $$1 + 684T + 3.00e5T^{2}$$
71 $$1 + 588T + 3.57e5T^{2}$$
73 $$1 + 270.T + 3.89e5T^{2}$$
79 $$1 + 1.22e3T + 4.93e5T^{2}$$
83 $$1 + 422.T + 5.71e5T^{2}$$
89 $$1 - 618.T + 7.04e5T^{2}$$
97 $$1 - 1.48e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$