Properties

 Label 2-768-24.5-c2-0-3 Degree $2$ Conductor $768$ Sign $-0.321 - 0.947i$ Analytic cond. $20.9264$ Root an. cond. $4.57454$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (1.32 − 2.69i)3-s − 0.640·5-s − 2.72·7-s + (−5.47 − 7.14i)9-s − 11.2·11-s + 5.25i·13-s + (−0.849 + 1.72i)15-s + 14.8i·17-s − 15.0i·19-s + (−3.61 + 7.31i)21-s + 36.4i·23-s − 24.5·25-s + (−26.4 + 5.24i)27-s + 51.7·29-s − 36.5·31-s + ⋯
 L(s)  = 1 + (0.442 − 0.896i)3-s − 0.128·5-s − 0.388·7-s + (−0.608 − 0.793i)9-s − 1.02·11-s + 0.403i·13-s + (−0.0566 + 0.114i)15-s + 0.874i·17-s − 0.793i·19-s + (−0.171 + 0.348i)21-s + 1.58i·23-s − 0.983·25-s + (−0.980 + 0.194i)27-s + 1.78·29-s − 1.17·31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $-0.321 - 0.947i$ Analytic conductor: $$20.9264$$ Root analytic conductor: $$4.57454$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{768} (641, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :1),\ -0.321 - 0.947i)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.4055489647$$ $$L(\frac12)$$ $$\approx$$ $$0.4055489647$$ $$L(2)$$ 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 + (-1.32 + 2.69i)T$$
good5 $$1 + 0.640T + 25T^{2}$$
7 $$1 + 2.72T + 49T^{2}$$
11 $$1 + 11.2T + 121T^{2}$$
13 $$1 - 5.25iT - 169T^{2}$$
17 $$1 - 14.8iT - 289T^{2}$$
19 $$1 + 15.0iT - 361T^{2}$$
23 $$1 - 36.4iT - 529T^{2}$$
29 $$1 - 51.7T + 841T^{2}$$
31 $$1 + 36.5T + 961T^{2}$$
37 $$1 - 63.6iT - 1.36e3T^{2}$$
41 $$1 - 12.1iT - 1.68e3T^{2}$$
43 $$1 + 11.8iT - 1.84e3T^{2}$$
47 $$1 + 61.1iT - 2.20e3T^{2}$$
53 $$1 + 59.1T + 2.80e3T^{2}$$
59 $$1 - 37.2T + 3.48e3T^{2}$$
61 $$1 - 58.1iT - 3.72e3T^{2}$$
67 $$1 + 23.0iT - 4.48e3T^{2}$$
71 $$1 - 7.29iT - 5.04e3T^{2}$$
73 $$1 + 73.4T + 5.32e3T^{2}$$
79 $$1 - 58.5T + 6.24e3T^{2}$$
83 $$1 + 32.3T + 6.88e3T^{2}$$
89 $$1 - 112. iT - 7.92e3T^{2}$$
97 $$1 + 80.0T + 9.40e3T^{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

−10.28404475944796385284520059425, −9.461583743181298908281383901685, −8.487368689241922979170533267722, −7.83977452184290471962159092137, −6.97136419846255732804531210914, −6.15987152212421922015738033139, −5.11380821141083434434738906603, −3.68842522703068876980457064251, −2.72879725638743395893496988269, −1.52919467581721838306730838760, 0.12440085083575591172986601742, 2.38408906353111682397969333231, 3.24289050493280379018933043935, 4.35954328010274066644907232309, 5.23151501497365374876260335367, 6.19026880842115633079816769404, 7.53672912506971642428253165002, 8.187210409447698844315384764057, 9.090879186706034491403024521736, 9.952773460382491965406249581721