Your ubuntu is almost certainly running on an Intel or AMD processor (and yes, it is 64-bits). The Zedboard uses a different processor, ARM, which has a completely different binary instruction format. So it is not only the 32 versus 64 bit, but the CPU architecture (ARM versus X86) that is different. Binaries compiled for one CPU will not run on the other, and vice-versa.
You cannot use the "gcc" on your host to generate binaries for ARM. Instead, you must use a cross-compiler. And for various reasons, you should use the same cross-compiler that was used to build all the other code on your ZedBoard. Otherwise you will likely have issues due to different C library, different ABI, different floating point options, etc.
Please have a look at Petalinux UG1144 as a starting point. If you don't want to use the whole petalinux-build process, then you can still make use of the eSDK that it provides, which will give you a cross-compiler for building ARM binaries on your X86 host. See "Installing a Preferred eSDK as part of the PetaLinux Tool" in the same UG1144 document. You will want the "arm" variant (this is 32-bit ARM), not "aarch64" (which is 64-bit ARM).