# Properties

 Label 2-76-19.7-c5-0-7 Degree $2$ Conductor $76$ Sign $-0.952 - 0.302i$ Analytic cond. $12.1891$ Root an. cond. $3.49129$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−6.79 − 11.7i)3-s + (−47.4 − 82.2i)5-s + 189.·7-s + (29.1 − 50.4i)9-s − 530.·11-s + (−353. + 612. i)13-s + (−645. + 1.11e3i)15-s + (−764. − 1.32e3i)17-s + (654. + 1.43e3i)19-s + (−1.29e3 − 2.23e3i)21-s + (497. − 861. i)23-s + (−2.94e3 + 5.10e3i)25-s − 4.09e3·27-s + (−1.29e3 + 2.23e3i)29-s − 2.79e3·31-s + ⋯
 L(s)  = 1 + (−0.435 − 0.755i)3-s + (−0.849 − 1.47i)5-s + 1.46·7-s + (0.119 − 0.207i)9-s − 1.32·11-s + (−0.580 + 1.00i)13-s + (−0.740 + 1.28i)15-s + (−0.641 − 1.11i)17-s + (0.415 + 0.909i)19-s + (−0.638 − 1.10i)21-s + (0.196 − 0.339i)23-s + (−0.943 + 1.63i)25-s − 1.08·27-s + (−0.284 + 0.493i)29-s − 0.521·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.302i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $-0.952 - 0.302i$ Analytic conductor: $$12.1891$$ Root analytic conductor: $$3.49129$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{76} (45, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :5/2),\ -0.952 - 0.302i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.108914 + 0.702079i$$ $$L(\frac12)$$ $$\approx$$ $$0.108914 + 0.702079i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + (-654. - 1.43e3i)T$$
good3 $$1 + (6.79 + 11.7i)T + (-121.5 + 210. i)T^{2}$$
5 $$1 + (47.4 + 82.2i)T + (-1.56e3 + 2.70e3i)T^{2}$$
7 $$1 - 189.T + 1.68e4T^{2}$$
11 $$1 + 530.T + 1.61e5T^{2}$$
13 $$1 + (353. - 612. i)T + (-1.85e5 - 3.21e5i)T^{2}$$
17 $$1 + (764. + 1.32e3i)T + (-7.09e5 + 1.22e6i)T^{2}$$
23 $$1 + (-497. + 861. i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + (1.29e3 - 2.23e3i)T + (-1.02e7 - 1.77e7i)T^{2}$$
31 $$1 + 2.79e3T + 2.86e7T^{2}$$
37 $$1 - 7.23e3T + 6.93e7T^{2}$$
41 $$1 + (-2.18e3 - 3.77e3i)T + (-5.79e7 + 1.00e8i)T^{2}$$
43 $$1 + (3.12e3 + 5.40e3i)T + (-7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (-1.23e4 + 2.13e4i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (1.54e4 - 2.68e4i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (1.81e4 + 3.14e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (-2.88e3 + 5.00e3i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (-3.03e4 + 5.26e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + (-1.48e4 - 2.56e4i)T + (-9.02e8 + 1.56e9i)T^{2}$$
73 $$1 + (3.90e4 + 6.75e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (3.34e4 + 5.78e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 2.19e4T + 3.93e9T^{2}$$
89 $$1 + (4.53e4 - 7.85e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 + (3.07e4 + 5.32e4i)T + (-4.29e9 + 7.43e9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$