# Properties

 Label 2-76-19.18-c6-0-1 Degree $2$ Conductor $76$ Sign $-0.635 + 0.772i$ 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 + 42.5i·3-s + 103.·5-s − 619.·7-s − 1.08e3·9-s − 57.6·11-s + 481. i·13-s + 4.40e3i·15-s − 2.33e3·17-s + (4.35e3 − 5.29e3i)19-s − 2.63e4i·21-s − 8.69e3·23-s − 4.92e3·25-s − 1.51e4i·27-s − 4.60e4i·29-s + 3.67e4i·31-s + ⋯
 L(s)  = 1 + 1.57i·3-s + 0.827·5-s − 1.80·7-s − 1.48·9-s − 0.0432·11-s + 0.219i·13-s + 1.30i·15-s − 0.476·17-s + (0.635 − 0.772i)19-s − 2.84i·21-s − 0.714·23-s − 0.315·25-s − 0.771i·27-s − 1.88i·29-s + 1.23i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $-0.635 + 0.772i$ 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: $$(2,\ 76,\ (\ :3),\ -0.635 + 0.772i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.168037 - 0.356017i$$ $$L(\frac12)$$ $$\approx$$ $$0.168037 - 0.356017i$$ $$L(4)$$ 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 + (-4.35e3 + 5.29e3i)T$$
good3 $$1 - 42.5iT - 729T^{2}$$
5 $$1 - 103.T + 1.56e4T^{2}$$
7 $$1 + 619.T + 1.17e5T^{2}$$
11 $$1 + 57.6T + 1.77e6T^{2}$$
13 $$1 - 481. iT - 4.82e6T^{2}$$
17 $$1 + 2.33e3T + 2.41e7T^{2}$$
23 $$1 + 8.69e3T + 1.48e8T^{2}$$
29 $$1 + 4.60e4iT - 5.94e8T^{2}$$
31 $$1 - 3.67e4iT - 8.87e8T^{2}$$
37 $$1 - 4.04e4iT - 2.56e9T^{2}$$
41 $$1 + 6.15e4iT - 4.75e9T^{2}$$
43 $$1 + 6.43e4T + 6.32e9T^{2}$$
47 $$1 + 9.39e4T + 1.07e10T^{2}$$
53 $$1 + 9.78e4iT - 2.21e10T^{2}$$
59 $$1 - 1.32e5iT - 4.21e10T^{2}$$
61 $$1 + 1.92e3T + 5.15e10T^{2}$$
67 $$1 - 3.78e5iT - 9.04e10T^{2}$$
71 $$1 - 4.51e5iT - 1.28e11T^{2}$$
73 $$1 + 6.69e5T + 1.51e11T^{2}$$
79 $$1 - 7.47e5iT - 2.43e11T^{2}$$
83 $$1 + 4.95e5T + 3.26e11T^{2}$$
89 $$1 - 1.35e6iT - 4.96e11T^{2}$$
97 $$1 + 7.59e4iT - 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$