# Properties

 Label 2-76-19.7-c5-0-6 Degree $2$ Conductor $76$ Sign $0.535 + 0.844i$ 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 + (4.20 + 7.28i)3-s + (−18.6 − 32.3i)5-s + 15.8·7-s + (86.1 − 149. i)9-s − 325.·11-s + (519. − 900. i)13-s + (156. − 271. i)15-s + (778. + 1.34e3i)17-s + (418. − 1.51e3i)19-s + (66.7 + 115. i)21-s + (784. − 1.35e3i)23-s + (866. − 1.50e3i)25-s + 3.49e3·27-s + (4.02e3 − 6.96e3i)29-s − 9.52e3·31-s + ⋯
 L(s)  = 1 + (0.269 + 0.467i)3-s + (−0.333 − 0.577i)5-s + 0.122·7-s + (0.354 − 0.614i)9-s − 0.811·11-s + (0.852 − 1.47i)13-s + (0.179 − 0.311i)15-s + (0.653 + 1.13i)17-s + (0.265 − 0.964i)19-s + (0.0330 + 0.0572i)21-s + (0.309 − 0.535i)23-s + (0.277 − 0.480i)25-s + 0.921·27-s + (0.888 − 1.53i)29-s − 1.78·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.535 + 0.844i$ 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: $$(2,\ 76,\ (\ :5/2),\ 0.535 + 0.844i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.50532 - 0.827660i$$ $$L(\frac12)$$ $$\approx$$ $$1.50532 - 0.827660i$$ $$L(\frac{7}{2})$$ 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 + (-418. + 1.51e3i)T$$
good3 $$1 + (-4.20 - 7.28i)T + (-121.5 + 210. i)T^{2}$$
5 $$1 + (18.6 + 32.3i)T + (-1.56e3 + 2.70e3i)T^{2}$$
7 $$1 - 15.8T + 1.68e4T^{2}$$
11 $$1 + 325.T + 1.61e5T^{2}$$
13 $$1 + (-519. + 900. i)T + (-1.85e5 - 3.21e5i)T^{2}$$
17 $$1 + (-778. - 1.34e3i)T + (-7.09e5 + 1.22e6i)T^{2}$$
23 $$1 + (-784. + 1.35e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + (-4.02e3 + 6.96e3i)T + (-1.02e7 - 1.77e7i)T^{2}$$
31 $$1 + 9.52e3T + 2.86e7T^{2}$$
37 $$1 + 451.T + 6.93e7T^{2}$$
41 $$1 + (-2.23e3 - 3.86e3i)T + (-5.79e7 + 1.00e8i)T^{2}$$
43 $$1 + (-5.77e3 - 1.00e4i)T + (-7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (3.08e3 - 5.33e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (1.11e4 - 1.93e4i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (3.51e3 + 6.08e3i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (-4.07e3 + 7.05e3i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (2.73e4 - 4.73e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + (-2.48e4 - 4.30e4i)T + (-9.02e8 + 1.56e9i)T^{2}$$
73 $$1 + (2.91e4 + 5.05e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (1.83e4 + 3.17e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 - 6.56e4T + 3.93e9T^{2}$$
89 $$1 + (2.01e4 - 3.48e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 + (5.67e3 + 9.83e3i)T + (-4.29e9 + 7.43e9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$