Custom 3rd person character controller for Unity

PlayerMovement.cs

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class PlayerMovement : MonoBehaviour {
    private CharacterController characterController;
    private Animator animator;
    private float moveSpeed;
    private bool hasJumped;

    [SerializeField]
    private float walkSpeed = 6;
    [SerializeField]
    private float runSpeed = 12;
    [SerializeField]
    private float turnSpeed = 50f;
    [SerializeField]
    private float jumpHeight = 50f;
    [SerializeField]
    private float jumpSpeed = 8.0f;
    [SerializeField]
    private float gravity = 9.8f;
    [SerializeField]
    private float vertical_speed = 0.0f;


    private Vector3 turnMovement;

    private void Awake()
    {
        characterController = GetComponent<CharacterController>();
        animator = GetComponentInChildren<Animator>();
    }

    private void Update()
    {

        /*
         * Animations that doesn't impact movement
         */
        if (Input.GetKeyDown("b"))
        {
            animator.SetTrigger("Blink");
        }
        if (Input.GetKeyDown("m"))
        {
            animator.SetTrigger("Meow");
        }


        /*
         * Movement
         */
        var movement = Vector3.zero;
        
        if (Input.GetKey("left shift"))
        {
            moveSpeed = runSpeed;
        }
        else
        {
            moveSpeed = walkSpeed;
        }

        var horizontal = Input.GetAxis("Horizontal");
        var vertical = Input.GetAxis("Vertical");

        movement = new Vector3(0, 0, vertical);

        Vector3 relativeMovement = transform.TransformVector(movement);
        relativeMovement *= moveSpeed;

    
        animator.SetFloat("Speed", movement.magnitude * moveSpeed);

        if (characterController.isGrounded)
        {
            animator.ResetTrigger("Falling");
            animator.ResetTrigger("Jump");
            hasJumped = false;
            vertical_speed = 0.0f;
            if (Input.GetKeyDown("space"))
            {
                hasJumped = true;
                animator.SetTrigger("Jump");
                relativeMovement.y = jumpHeight;
                vertical_speed = jumpSpeed;
            }

        }
        else
        {
            if (!hasJumped && vertical_speed < -3.0f)
            {
                animator.SetTrigger("Falling");
            }
            
        }

        Debug.Log("Vertical speed: " + vertical_speed);

        vertical_speed -= gravity * Time.deltaTime;

        relativeMovement.y = vertical_speed;
        
        characterController.Move(relativeMovement * Time.deltaTime);
        transform.Rotate(Vector3.up, horizontal * turnSpeed * Time.deltaTime);

    }

}