# Properties

 Label 2-76-19.14-c4-0-1 Degree $2$ Conductor $76$ Sign $0.661 - 0.749i$ Analytic cond. $7.85611$ Root an. cond. $2.80287$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−17.2 − 3.03i)3-s + (−15.3 − 12.9i)5-s + (−26.4 − 45.7i)7-s + (211. + 76.8i)9-s + (−106. + 184. i)11-s + (67.2 − 11.8i)13-s + (225. + 269. i)15-s + (86.5 − 31.4i)17-s + (360. − 10.5i)19-s + (316. + 868. i)21-s + (152. − 128. i)23-s + (−38.3 − 217. i)25-s + (−2.17e3 − 1.25e3i)27-s + (−125. + 345. i)29-s + (−715. + 412. i)31-s + ⋯
 L(s)  = 1 + (−1.91 − 0.337i)3-s + (−0.615 − 0.516i)5-s + (−0.539 − 0.934i)7-s + (2.60 + 0.948i)9-s + (−0.880 + 1.52i)11-s + (0.398 − 0.0702i)13-s + (1.00 + 1.19i)15-s + (0.299 − 0.108i)17-s + (0.999 − 0.0293i)19-s + (0.716 + 1.96i)21-s + (0.288 − 0.242i)23-s + (−0.0614 − 0.348i)25-s + (−2.98 − 1.72i)27-s + (−0.149 + 0.411i)29-s + (−0.744 + 0.429i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.661 - 0.749i$ Analytic conductor: $$7.85611$$ Root analytic conductor: $$2.80287$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{76} (33, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :2),\ 0.661 - 0.749i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.396345 + 0.178878i$$ $$L(\frac12)$$ $$\approx$$ $$0.396345 + 0.178878i$$ $$L(3)$$ 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 + (-360. + 10.5i)T$$
good3 $$1 + (17.2 + 3.03i)T + (76.1 + 27.7i)T^{2}$$
5 $$1 + (15.3 + 12.9i)T + (108. + 615. i)T^{2}$$
7 $$1 + (26.4 + 45.7i)T + (-1.20e3 + 2.07e3i)T^{2}$$
11 $$1 + (106. - 184. i)T + (-7.32e3 - 1.26e4i)T^{2}$$
13 $$1 + (-67.2 + 11.8i)T + (2.68e4 - 9.76e3i)T^{2}$$
17 $$1 + (-86.5 + 31.4i)T + (6.39e4 - 5.36e4i)T^{2}$$
23 $$1 + (-152. + 128. i)T + (4.85e4 - 2.75e5i)T^{2}$$
29 $$1 + (125. - 345. i)T + (-5.41e5 - 4.54e5i)T^{2}$$
31 $$1 + (715. - 412. i)T + (4.61e5 - 7.99e5i)T^{2}$$
37 $$1 - 2.46e3iT - 1.87e6T^{2}$$
41 $$1 + (-1.69e3 - 299. i)T + (2.65e6 + 9.66e5i)T^{2}$$
43 $$1 + (1.34e3 + 1.12e3i)T + (5.93e5 + 3.36e6i)T^{2}$$
47 $$1 + (-1.84e3 - 670. i)T + (3.73e6 + 3.13e6i)T^{2}$$
53 $$1 + (-1.28e3 - 1.52e3i)T + (-1.37e6 + 7.77e6i)T^{2}$$
59 $$1 + (-836. - 2.29e3i)T + (-9.28e6 + 7.78e6i)T^{2}$$
61 $$1 + (1.06e3 - 891. i)T + (2.40e6 - 1.36e7i)T^{2}$$
67 $$1 + (-415. + 1.14e3i)T + (-1.54e7 - 1.29e7i)T^{2}$$
71 $$1 + (5.93e3 - 7.07e3i)T + (-4.41e6 - 2.50e7i)T^{2}$$
73 $$1 + (258. - 1.46e3i)T + (-2.66e7 - 9.71e6i)T^{2}$$
79 $$1 + (-8.75e3 - 1.54e3i)T + (3.66e7 + 1.33e7i)T^{2}$$
83 $$1 + (-3.71e3 - 6.44e3i)T + (-2.37e7 + 4.11e7i)T^{2}$$
89 $$1 + (4.04e3 - 712. i)T + (5.89e7 - 2.14e7i)T^{2}$$
97 $$1 + (2.46e3 + 6.77e3i)T + (-6.78e7 + 5.69e7i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$