# Properties

 Label 2-76-76.7-c2-0-12 Degree $2$ Conductor $76$ Sign $0.287 + 0.957i$ Analytic cond. $2.07085$ Root an. cond. $1.43904$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.99 + 0.147i)2-s + (3.58 − 2.06i)3-s + (3.95 − 0.589i)4-s + (−3.72 − 6.46i)5-s + (−6.83 + 4.65i)6-s + 3.06i·7-s + (−7.80 + 1.76i)8-s + (4.05 − 7.02i)9-s + (8.39 + 12.3i)10-s − 6.31i·11-s + (12.9 − 10.2i)12-s + (8.74 − 15.1i)13-s + (−0.453 − 6.12i)14-s + (−26.7 − 15.4i)15-s + (15.3 − 4.66i)16-s + (10.6 + 18.4i)17-s + ⋯
 L(s)  = 1 + (−0.997 + 0.0738i)2-s + (1.19 − 0.689i)3-s + (0.989 − 0.147i)4-s + (−0.745 − 1.29i)5-s + (−1.13 + 0.775i)6-s + 0.438i·7-s + (−0.975 + 0.220i)8-s + (0.450 − 0.780i)9-s + (0.839 + 1.23i)10-s − 0.574i·11-s + (1.07 − 0.857i)12-s + (0.672 − 1.16i)13-s + (−0.0323 − 0.437i)14-s + (−1.78 − 1.02i)15-s + (0.956 − 0.291i)16-s + (0.628 + 1.08i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.287 + 0.957i$ Analytic conductor: $$2.07085$$ Root analytic conductor: $$1.43904$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{76} (7, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :1),\ 0.287 + 0.957i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.852558 - 0.634440i$$ $$L(\frac12)$$ $$\approx$$ $$0.852558 - 0.634440i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.99 - 0.147i)T$$
19 $$1 + (-6.62 - 17.8i)T$$
good3 $$1 + (-3.58 + 2.06i)T + (4.5 - 7.79i)T^{2}$$
5 $$1 + (3.72 + 6.46i)T + (-12.5 + 21.6i)T^{2}$$
7 $$1 - 3.06iT - 49T^{2}$$
11 $$1 + 6.31iT - 121T^{2}$$
13 $$1 + (-8.74 + 15.1i)T + (-84.5 - 146. i)T^{2}$$
17 $$1 + (-10.6 - 18.4i)T + (-144.5 + 250. i)T^{2}$$
23 $$1 + (5.19 + 3.00i)T + (264.5 + 458. i)T^{2}$$
29 $$1 + (-18.2 + 31.6i)T + (-420.5 - 728. i)T^{2}$$
31 $$1 - 53.3iT - 961T^{2}$$
37 $$1 + 39.2T + 1.36e3T^{2}$$
41 $$1 + (-18.2 - 31.6i)T + (-840.5 + 1.45e3i)T^{2}$$
43 $$1 + (-31.7 + 18.3i)T + (924.5 - 1.60e3i)T^{2}$$
47 $$1 + (-0.0577 - 0.0333i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 + (31.6 - 54.8i)T + (-1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (41.8 - 24.1i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (-29.8 + 51.7i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (58.4 + 33.7i)T + (2.24e3 + 3.88e3i)T^{2}$$
71 $$1 + (-30.0 + 17.3i)T + (2.52e3 - 4.36e3i)T^{2}$$
73 $$1 + (17.7 + 30.7i)T + (-2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (65.0 - 37.5i)T + (3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + 53.5iT - 6.88e3T^{2}$$
89 $$1 + (-36.5 + 63.2i)T + (-3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + (59.7 + 103. i)T + (-4.70e3 + 8.14e3i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$