# Properties

 Label 2-76-19.18-c6-0-0 Degree $2$ Conductor $76$ Sign $0.147 - 0.989i$ Analytic cond. $17.4841$ Root an. cond. $4.18140$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 40.3i·3-s − 186.·5-s + 202.·7-s − 898.·9-s − 1.59e3·11-s − 143. i·13-s + 7.53e3i·15-s + 4.77e3·17-s + (−1.01e3 + 6.78e3i)19-s − 8.16e3i·21-s + 7.73e3·23-s + 1.92e4·25-s + 6.83e3i·27-s − 8.82e3i·29-s + 5.15e4i·31-s + ⋯
 L(s)  = 1 − 1.49i·3-s − 1.49·5-s + 0.589·7-s − 1.23·9-s − 1.19·11-s − 0.0652i·13-s + 2.23i·15-s + 0.971·17-s + (−0.147 + 0.989i)19-s − 0.881i·21-s + 0.635·23-s + 1.23·25-s + 0.347i·27-s − 0.361i·29-s + 1.73i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.147 - 0.989i$ 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: $$(2,\ 76,\ (\ :3),\ 0.147 - 0.989i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.187357 + 0.161518i$$ $$L(\frac12)$$ $$\approx$$ $$0.187357 + 0.161518i$$ $$L(4)$$ 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 + (1.01e3 - 6.78e3i)T$$
good3 $$1 + 40.3iT - 729T^{2}$$
5 $$1 + 186.T + 1.56e4T^{2}$$
7 $$1 - 202.T + 1.17e5T^{2}$$
11 $$1 + 1.59e3T + 1.77e6T^{2}$$
13 $$1 + 143. iT - 4.82e6T^{2}$$
17 $$1 - 4.77e3T + 2.41e7T^{2}$$
23 $$1 - 7.73e3T + 1.48e8T^{2}$$
29 $$1 + 8.82e3iT - 5.94e8T^{2}$$
31 $$1 - 5.15e4iT - 8.87e8T^{2}$$
37 $$1 - 9.46e4iT - 2.56e9T^{2}$$
41 $$1 + 7.87e4iT - 4.75e9T^{2}$$
43 $$1 + 1.29e5T + 6.32e9T^{2}$$
47 $$1 + 9.72e4T + 1.07e10T^{2}$$
53 $$1 - 4.61e4iT - 2.21e10T^{2}$$
59 $$1 + 6.17e4iT - 4.21e10T^{2}$$
61 $$1 + 8.56e4T + 5.15e10T^{2}$$
67 $$1 - 4.90e4iT - 9.04e10T^{2}$$
71 $$1 + 4.08e5iT - 1.28e11T^{2}$$
73 $$1 + 1.30e5T + 1.51e11T^{2}$$
79 $$1 - 1.06e5iT - 2.43e11T^{2}$$
83 $$1 + 1.22e4T + 3.26e11T^{2}$$
89 $$1 - 5.61e5iT - 4.96e11T^{2}$$
97 $$1 - 8.51e5iT - 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

−13.33433679934479817290468659184, −12.30864949331904822117759011145, −11.76380754910992687093295778237, −10.50154751867358037301088884852, −8.158683876762347505190363806333, −7.961642605726728131620105117031, −6.81579948605464028420962297232, −5.08627815986691194357724860715, −3.19618052833142988101970536198, −1.37457836824521052146153700663, 0.099918861017052233589202121178, 3.14135421653420517638554082514, 4.33644624358380704148085202061, 5.22457789001735216481205751997, 7.51353674409338609420878058380, 8.445897983097858856999705937342, 9.785941567124836651944950121113, 10.96172415365668557215206806695, 11.49515246172940547217160330113, 12.96353956266022525185840016262