# Properties

 Label 2-768-1.1-c3-0-12 Degree $2$ Conductor $768$ Sign $1$ Analytic cond. $45.3134$ Root an. cond. $6.73152$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s + 3.46·5-s + 24.2·7-s + 9·9-s − 48·11-s + 41.5·13-s − 10.3·15-s + 54·17-s + 4·19-s − 72.7·21-s + 173.·23-s − 113·25-s − 27·27-s + 162.·29-s − 58.8·31-s + 144·33-s + 84·35-s − 325.·37-s − 124.·39-s + 294·41-s − 188·43-s + 31.1·45-s − 505.·47-s + 245·49-s − 162·51-s + 744.·53-s − 166.·55-s + ⋯
 L(s)  = 1 − 0.577·3-s + 0.309·5-s + 1.30·7-s + 0.333·9-s − 1.31·11-s + 0.886·13-s − 0.178·15-s + 0.770·17-s + 0.0482·19-s − 0.755·21-s + 1.57·23-s − 0.904·25-s − 0.192·27-s + 1.04·29-s − 0.341·31-s + 0.759·33-s + 0.405·35-s − 1.44·37-s − 0.512·39-s + 1.11·41-s − 0.666·43-s + 0.103·45-s − 1.56·47-s + 0.714·49-s − 0.444·51-s + 1.93·53-s − 0.407·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $1$ Analytic conductor: $$45.3134$$ Root analytic conductor: $$6.73152$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.089822593$$ $$L(\frac12)$$ $$\approx$$ $$2.089822593$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + 3T$$
good5 $$1 - 3.46T + 125T^{2}$$
7 $$1 - 24.2T + 343T^{2}$$
11 $$1 + 48T + 1.33e3T^{2}$$
13 $$1 - 41.5T + 2.19e3T^{2}$$
17 $$1 - 54T + 4.91e3T^{2}$$
19 $$1 - 4T + 6.85e3T^{2}$$
23 $$1 - 173.T + 1.21e4T^{2}$$
29 $$1 - 162.T + 2.43e4T^{2}$$
31 $$1 + 58.8T + 2.97e4T^{2}$$
37 $$1 + 325.T + 5.06e4T^{2}$$
41 $$1 - 294T + 6.89e4T^{2}$$
43 $$1 + 188T + 7.95e4T^{2}$$
47 $$1 + 505.T + 1.03e5T^{2}$$
53 $$1 - 744.T + 1.48e5T^{2}$$
59 $$1 + 252T + 2.05e5T^{2}$$
61 $$1 + 90.0T + 2.26e5T^{2}$$
67 $$1 + 628T + 3.00e5T^{2}$$
71 $$1 + 6.92T + 3.57e5T^{2}$$
73 $$1 - 1.00e3T + 3.89e5T^{2}$$
79 $$1 - 1.34e3T + 4.93e5T^{2}$$
83 $$1 - 720T + 5.71e5T^{2}$$
89 $$1 - 1.48e3T + 7.04e5T^{2}$$
97 $$1 - 1.82e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$