We propose a shrinkage method to estimate the coefficient function in a functional linear regression model when the value of the coefficient function is zero within certain sub-regions. The primary goal is to perform estimation and inferences for the nonparametric estimate of the coefficient function and to identify the null region in which the function is zero. Our proposal consists of two stages. In stage one, Dantzig selector is employed to provide initial location of the null region. In stage two, we propose a group SCAD approach to refine the estimated location of the null region and to provide the estimation and inference procedures for the coefficient function. Our grouping consideration has certain advantageous contributions in this functional setup. Theoretically, we show that our estimator enjoys the desired Oracle property; it identifies the null region with probability tending to 1, and it achieves the same asymptotic normality for the estimated coefficient function on the non-null region, as in the regular functional linear model estimation when the non-null region is known. Numerically, owing to the additional refinement stage, we only need to use a large but not an extremely large number of knots in both stages, yet achieve superior numerical performances. This simple practice also ensures that the method can be easily used by practitioners without the concern of skillfully handling an exceptionally large number of parameters that might be required in other approaches. Our refined estimator overcomes the shortcomings of the initial Dantzig estimator which tends to over-estimate the null region and under-estimate the absolute scale of non-zero coefficients. The numerical performances of the proposed method are illustrated in a simulation study. They are also shown in an analysis of data collected by the Johns Hopkins Precursors Study where the primary interests are estimating the strength of association between body mass index in midlife and the quality of life in physical functioning at old age, and identifying the effective age ranges where such associations exist.