# Properties

 Label 2-76-19.11-c5-0-0 Degree $2$ Conductor $76$ Sign $-0.104 + 0.994i$ 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 + (−11.6 + 20.2i)3-s + (−25.2 + 43.7i)5-s − 187.·7-s + (−150. − 261. i)9-s + 411.·11-s + (286. + 496. i)13-s + (−589. − 1.02e3i)15-s + (559. − 968. i)17-s + (−1.04e3 − 1.17e3i)19-s + (2.19e3 − 3.79e3i)21-s + (−72.0 − 124. i)23-s + (287. + 498. i)25-s + 1.36e3·27-s + (−2.61e3 − 4.53e3i)29-s − 6.72e3·31-s + ⋯
 L(s)  = 1 + (−0.748 + 1.29i)3-s + (−0.451 + 0.782i)5-s − 1.44·7-s + (−0.620 − 1.07i)9-s + 1.02·11-s + (0.470 + 0.815i)13-s + (−0.676 − 1.17i)15-s + (0.469 − 0.812i)17-s + (−0.666 − 0.745i)19-s + (1.08 − 1.87i)21-s + (−0.0283 − 0.0491i)23-s + (0.0920 + 0.159i)25-s + 0.361·27-s + (−0.577 − 1.00i)29-s − 1.25·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $-0.104 + 0.994i$ Analytic conductor: $$12.1891$$ Root analytic conductor: $$3.49129$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{76} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :5/2),\ -0.104 + 0.994i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.0611710 - 0.0679336i$$ $$L(\frac12)$$ $$\approx$$ $$0.0611710 - 0.0679336i$$ $$L(\frac{7}{2})$$ 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 + (1.04e3 + 1.17e3i)T$$
good3 $$1 + (11.6 - 20.2i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (25.2 - 43.7i)T + (-1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + 187.T + 1.68e4T^{2}$$
11 $$1 - 411.T + 1.61e5T^{2}$$
13 $$1 + (-286. - 496. i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 + (-559. + 968. i)T + (-7.09e5 - 1.22e6i)T^{2}$$
23 $$1 + (72.0 + 124. i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + (2.61e3 + 4.53e3i)T + (-1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + 6.72e3T + 2.86e7T^{2}$$
37 $$1 - 1.23e4T + 6.93e7T^{2}$$
41 $$1 + (5.96e3 - 1.03e4i)T + (-5.79e7 - 1.00e8i)T^{2}$$
43 $$1 + (2.61e3 - 4.52e3i)T + (-7.35e7 - 1.27e8i)T^{2}$$
47 $$1 + (1.04e4 + 1.81e4i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (-4.55e3 - 7.88e3i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 + (1.20e4 - 2.08e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (2.15e4 + 3.73e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (8.05e3 + 1.39e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 + (3.99e4 - 6.91e4i)T + (-9.02e8 - 1.56e9i)T^{2}$$
73 $$1 + (1.72e4 - 2.98e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (-1.63e4 + 2.83e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 - 4.96e4T + 3.93e9T^{2}$$
89 $$1 + (3.92e4 + 6.79e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 + (-6.26e4 + 1.08e5i)T + (-4.29e9 - 7.43e9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$