# Properties

 Label 2-76-19.18-c6-0-6 Degree $2$ Conductor $76$ Sign $0.398 - 0.916i$ Analytic cond. $17.4841$ Root an. cond. $4.18140$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 38.1i·3-s + 102.·5-s + 512.·7-s − 726.·9-s + 2.41e3·11-s − 3.77e3i·13-s + 3.90e3i·15-s + 1.56e3·17-s + (−2.73e3 + 6.28e3i)19-s + 1.95e4i·21-s − 626.·23-s − 5.12e3·25-s + 96.4i·27-s + 1.57e4i·29-s − 2.61e4i·31-s + ⋯
 L(s)  = 1 + 1.41i·3-s + 0.819·5-s + 1.49·7-s − 0.996·9-s + 1.81·11-s − 1.71i·13-s + 1.15i·15-s + 0.318·17-s + (−0.398 + 0.916i)19-s + 2.11i·21-s − 0.0514·23-s − 0.327·25-s + 0.00489i·27-s + 0.643i·29-s − 0.879i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.916i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.398 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.398 - 0.916i$ Analytic conductor: $$17.4841$$ Root analytic conductor: $$4.18140$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{76} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :3),\ 0.398 - 0.916i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$2.26934 + 1.48750i$$ $$L(\frac12)$$ $$\approx$$ $$2.26934 + 1.48750i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + (2.73e3 - 6.28e3i)T$$
good3 $$1 - 38.1iT - 729T^{2}$$
5 $$1 - 102.T + 1.56e4T^{2}$$
7 $$1 - 512.T + 1.17e5T^{2}$$
11 $$1 - 2.41e3T + 1.77e6T^{2}$$
13 $$1 + 3.77e3iT - 4.82e6T^{2}$$
17 $$1 - 1.56e3T + 2.41e7T^{2}$$
23 $$1 + 626.T + 1.48e8T^{2}$$
29 $$1 - 1.57e4iT - 5.94e8T^{2}$$
31 $$1 + 2.61e4iT - 8.87e8T^{2}$$
37 $$1 - 4.94e4iT - 2.56e9T^{2}$$
41 $$1 + 1.19e5iT - 4.75e9T^{2}$$
43 $$1 + 1.33e4T + 6.32e9T^{2}$$
47 $$1 + 1.86e5T + 1.07e10T^{2}$$
53 $$1 - 2.04e5iT - 2.21e10T^{2}$$
59 $$1 + 1.82e4iT - 4.21e10T^{2}$$
61 $$1 + 3.53e5T + 5.15e10T^{2}$$
67 $$1 + 5.19e5iT - 9.04e10T^{2}$$
71 $$1 + 2.68e5iT - 1.28e11T^{2}$$
73 $$1 - 5.85e4T + 1.51e11T^{2}$$
79 $$1 - 4.73e5iT - 2.43e11T^{2}$$
83 $$1 - 1.27e5T + 3.26e11T^{2}$$
89 $$1 - 6.07e5iT - 4.96e11T^{2}$$
97 $$1 + 2.82e5iT - 8.32e11T^{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

−13.90326215620337080595992074055, −12.19318625533498406139055447936, −11.00241644448962703920474202194, −10.16560652443244514593526788983, −9.178763416242989429124200326912, −8.008859832895877256123297594320, −5.94072944848448117514791268820, −4.87917508834811940726653087756, −3.63653121635304310063833698014, −1.52203592176866892054030876066, 1.38481190006585617389101392727, 1.90481307981278954736588639204, 4.49790604896557482901876417388, 6.25119215300870954724100826257, 7.03246730815842791717160387625, 8.407286572326488424995337547529, 9.453854943780647633886822945444, 11.45034277358976109611665016865, 11.80317066852144588193323930476, 13.23423979998528941774082824942