Imported raster won't align with existing shapefile - crs is the same

[MarionFrotzbacher]

I'm trying to make a raster layer with climate data align to my shapefiles I'm working with, but no matter what I try it won't work. The raster is projected correctly but far away from the shapefiles and has different coordinates. 

When importing the raster layer from an ASCII file it's displayed correctly but the location is wrong. I tried the Project Raster and the Define projection Tool, but neither worked.

I attach the info of the layers and a screenshot of the distance.

Many thanks in advance!

[JakeSkinner]

Hi Marion,

Even though the raster dataset has a coordinate system, it may still need to be georeferenced.  See the following document that describes how to do this:

Fundamentals of georeferencing a raster dataset—Help | ArcGIS for Desktop