How to calculate the following integral

$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$

where $\langle z, \zeta \rangle $ is the inner product in $ \mathbb{C}^{n} , z \in \mathbb{C}^{n}$ is a fixed vector, $ \zeta $ is a vector on the unit sphere $ S_{2n-1}$ and $ a $ are constants, and $ d\sigma(\zeta) $ is the surface measure on the sphere. Thank you in advance

How to (analytically) calculate the following integral,

$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$

where $\langle z, \zeta \rangle $ is the inner product in $ \mathbb{C}^{n} , z \in \mathbb{C}^{n}$ is a fixed vector, $ a,b $ are constants, $ \zeta $ is a vector on the unit sphere $ S_{2n-1}$, and $ d\sigma(\zeta) $ is the surface measure on the sphere.

How to evaluate the following integral

$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$

where $\langle z, \zeta \rangle $ is the inner product in $ \mathbb{C}^{n} , z \in \mathbb{C}^{n}$ is a fixed vector, $ \zeta $ is a vector on the unit sphere $ S_{2n-1}$ and $ a $ are constants, and $ d\sigma(\zeta) $ is the surface measure on the sphere. Thank you in advance

How to calculate the following integral

$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$

where $\langle z, \zeta \rangle $ is the inner product in $ \mathbb{C}^{n} , z \in \mathbb{C}^{n}$ is a fixed vector, $ \zeta $ is a vector on the unit sphere $ S_{2n-1}$ and $ a $ are constants, and $ d\sigma(\zeta) $ is the surface measure on the sphere. Thank you in advance