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Is the value of the following integral over the unitary group with respect to the normalized Haar measure known? $$ \int |Tr(U)|^pdU. $$ There are some results for $p=2k$, $k\in\mathbb N$ (see Diaconis and Evans, 'Linear functionals of eigenvalues of random matrices', Trans.of AMS, 2001).

Is the case $p=1$ known?

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    $\begingroup$ In the limit (as the dimension of the matrix gets larger) this should simply tend to the $p$th absolute moment of a standard complex Gaussian, by the work of Diaconis-Shahshahani. The convergence is superexponential, by work of Johansson. I wouldn't expect a closed-form evaluation for non-even moments, unless some Selberg integral miracle happens. $\endgroup$
    – Ofir Gorodetsky
    Commented Oct 18, 2021 at 19:31

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Here is a plot of the average of $|{\rm Tr}\,U|$ over ${\rm U}(n)$ as a function of the matrix size $n/10$ (averaged over $4\cdot10^3$ random matrices). The horizontal line is the large-$n$ asymptote noted by Ofir Gorodetsky, $$\int_{{\rm U}(n)} |\,{\rm Tr}\,(U)|\,dU\rightarrow \frac{1}{\pi}\int_{-\infty}^\infty dx\int_{-\infty}^\infty dy\,\sqrt{x^2+y^2} e^{-x^2-y^2}= \tfrac{1}{2}\sqrt\pi,$$ which is reached within one percent for $n\gtrsim 10$.

horizontal axis is matrix size $n$ divided by 10

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