# Properties

 Label 2-768-1.1-c3-0-41 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s + 9.15·5-s + 27.4·7-s + 9·9-s − 20.5·11-s − 32.0·13-s − 27.4·15-s − 111.·17-s − 129.·19-s − 82.2·21-s + 9.16·23-s − 41.1·25-s − 27·27-s − 41.0·29-s + 187.·31-s + 61.5·33-s + 251.·35-s + 114.·37-s + 96.1·39-s − 282.·41-s + 89.3·43-s + 82.3·45-s + 54.6·47-s + 408.·49-s + 335.·51-s − 726.·53-s − 187.·55-s + ⋯
 L(s)  = 1 − 0.577·3-s + 0.818·5-s + 1.48·7-s + 0.333·9-s − 0.562·11-s − 0.683·13-s − 0.472·15-s − 1.59·17-s − 1.56·19-s − 0.854·21-s + 0.0830·23-s − 0.329·25-s − 0.192·27-s − 0.262·29-s + 1.08·31-s + 0.324·33-s + 1.21·35-s + 0.507·37-s + 0.394·39-s − 1.07·41-s + 0.317·43-s + 0.272·45-s + 0.169·47-s + 1.19·49-s + 0.920·51-s − 1.88·53-s − 0.460·55-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: $$1$$ Selberg data: $$(2,\ 768,\ (\ :3/2),\ -1)$$

## 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$$
3 $$1 + 3T$$
good5 $$1 - 9.15T + 125T^{2}$$
7 $$1 - 27.4T + 343T^{2}$$
11 $$1 + 20.5T + 1.33e3T^{2}$$
13 $$1 + 32.0T + 2.19e3T^{2}$$
17 $$1 + 111.T + 4.91e3T^{2}$$
19 $$1 + 129.T + 6.85e3T^{2}$$
23 $$1 - 9.16T + 1.21e4T^{2}$$
29 $$1 + 41.0T + 2.43e4T^{2}$$
31 $$1 - 187.T + 2.97e4T^{2}$$
37 $$1 - 114.T + 5.06e4T^{2}$$
41 $$1 + 282.T + 6.89e4T^{2}$$
43 $$1 - 89.3T + 7.95e4T^{2}$$
47 $$1 - 54.6T + 1.03e5T^{2}$$
53 $$1 + 726.T + 1.48e5T^{2}$$
59 $$1 - 216.T + 2.05e5T^{2}$$
61 $$1 + 754.T + 2.26e5T^{2}$$
67 $$1 + 379.T + 3.00e5T^{2}$$
71 $$1 - 302.T + 3.57e5T^{2}$$
73 $$1 - 504.T + 3.89e5T^{2}$$
79 $$1 + 301.T + 4.93e5T^{2}$$
83 $$1 + 599.T + 5.71e5T^{2}$$
89 $$1 - 277.T + 7.04e5T^{2}$$
97 $$1 + 765.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$