# Properties

 Label 2-76-4.3-c2-0-2 Degree $2$ Conductor $76$ Sign $0.529 - 0.848i$ 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 + (−1.92 − 0.551i)2-s + 0.644i·3-s + (3.39 + 2.11i)4-s − 2.32·5-s + (0.355 − 1.23i)6-s + 8.62i·7-s + (−5.35 − 5.94i)8-s + 8.58·9-s + (4.47 + 1.28i)10-s + 19.2i·11-s + (−1.36 + 2.18i)12-s + 13.8·13-s + (4.75 − 16.5i)14-s − 1.50i·15-s + (7.01 + 14.3i)16-s − 12.5·17-s + ⋯
 L(s)  = 1 + (−0.961 − 0.275i)2-s + 0.214i·3-s + (0.848 + 0.529i)4-s − 0.465·5-s + (0.0592 − 0.206i)6-s + 1.23i·7-s + (−0.669 − 0.743i)8-s + 0.953·9-s + (0.447 + 0.128i)10-s + 1.75i·11-s + (−0.113 + 0.182i)12-s + 1.06·13-s + (0.339 − 1.18i)14-s − 0.100i·15-s + (0.438 + 0.898i)16-s − 0.736·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.529 - 0.848i$ Analytic conductor: $$2.07085$$ Root analytic conductor: $$1.43904$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{76} (39, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :1),\ 0.529 - 0.848i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.685531 + 0.380019i$$ $$L(\frac12)$$ $$\approx$$ $$0.685531 + 0.380019i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.92 + 0.551i)T$$
19 $$1 - 4.35iT$$
good3 $$1 - 0.644iT - 9T^{2}$$
5 $$1 + 2.32T + 25T^{2}$$
7 $$1 - 8.62iT - 49T^{2}$$
11 $$1 - 19.2iT - 121T^{2}$$
13 $$1 - 13.8T + 169T^{2}$$
17 $$1 + 12.5T + 289T^{2}$$
23 $$1 + 37.4iT - 529T^{2}$$
29 $$1 + 6.36T + 841T^{2}$$
31 $$1 + 5.44iT - 961T^{2}$$
37 $$1 - 20.9T + 1.36e3T^{2}$$
41 $$1 - 72.8T + 1.68e3T^{2}$$
43 $$1 - 10.1iT - 1.84e3T^{2}$$
47 $$1 + 32.4iT - 2.20e3T^{2}$$
53 $$1 + 42.9T + 2.80e3T^{2}$$
59 $$1 - 38.9iT - 3.48e3T^{2}$$
61 $$1 - 25.5T + 3.72e3T^{2}$$
67 $$1 + 65.3iT - 4.48e3T^{2}$$
71 $$1 + 18.7iT - 5.04e3T^{2}$$
73 $$1 - 72.8T + 5.32e3T^{2}$$
79 $$1 + 139. iT - 6.24e3T^{2}$$
83 $$1 - 94.7iT - 6.88e3T^{2}$$
89 $$1 - 33.3T + 7.92e3T^{2}$$
97 $$1 + 150.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$