# Properties

 Label 2-768-12.11-c1-0-16 Degree $2$ Conductor $768$ Sign $1$ Analytic cond. $6.13251$ Root an. cond. $2.47639$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.73·3-s − 2.82i·5-s + 4.89i·7-s + 2.99·9-s + 3.46·11-s − 4.89i·15-s + 8.48i·21-s − 3.00·25-s + 5.19·27-s + 2.82i·29-s − 4.89i·31-s + 5.99·33-s + 13.8·35-s − 8.48i·45-s − 16.9·49-s + ⋯
 L(s)  = 1 + 1.00·3-s − 1.26i·5-s + 1.85i·7-s + 0.999·9-s + 1.04·11-s − 1.26i·15-s + 1.85i·21-s − 0.600·25-s + 1.00·27-s + 0.525i·29-s − 0.879i·31-s + 1.04·33-s + 2.34·35-s − 1.26i·45-s − 2.42·49-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $1$ Analytic conductor: $$6.13251$$ Root analytic conductor: $$2.47639$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{768} (767, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.31831$$ $$L(\frac12)$$ $$\approx$$ $$2.31831$$ $$L(\frac{3}{2})$$ 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 - 1.73T$$
good5 $$1 + 2.82iT - 5T^{2}$$
7 $$1 - 4.89iT - 7T^{2}$$
11 $$1 - 3.46T + 11T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 2.82iT - 29T^{2}$$
31 $$1 + 4.89iT - 31T^{2}$$
37 $$1 + 37T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 14.1iT - 53T^{2}$$
59 $$1 - 10.3T + 59T^{2}$$
61 $$1 + 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 14T + 73T^{2}$$
79 $$1 - 14.6iT - 79T^{2}$$
83 $$1 + 17.3T + 83T^{2}$$
89 $$1 - 89T^{2}$$
97 $$1 - 2T + 97T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.812357515712544434123705176143, −9.279583949511853532075757120473, −8.616706418367673153960466053785, −8.257482664673325897802758815756, −6.88286165107540979374100367524, −5.76290138329858101247113134895, −4.89664487839473580130220161652, −3.83292364910677482169371624067, −2.54947405039769471395279578509, −1.49911110788580926479746368750, 1.36425066558286993075187059679, 2.89904344157900136338280407218, 3.75655952466733903797859199920, 4.41892135810656244908119157414, 6.33069846030775356871813541348, 7.11533425774731656430088601478, 7.45213944983189682014563506023, 8.578577146622862968058196661293, 9.629837680480124720927457136546, 10.32909760570079221481890994929