Evidence of a sharp multi-modal distribution in the relative position of the highest Riemann Siegel Z function analogue peaks of 5 periodic 1st degree L-functions and Davenport Heilbronn functions, along the critical line in the piecewise intervals [N^2 · 2π/5 , (N + 1)^2 · 2π/5 ), N ∈ Z.

Analysis of the distribution of the extreme Riemann Siegel Z function peaks on the critical line ($s=1/2+I\cdot t$) for the two 5-periodic 1st degree Dirichlet L-functions $L(\chi_5(3,.),s)$, $L(\chi_5(2,.),s)$ and their Davenport-Heilbronn function counterparts $f_{1}(s)$ and $f_{2}(s)$ with the most extreme peak in each piecewise interval $[N^2\cdot\frac{2\pi}{5},(N+1)^2\cdot\frac{2\pi}{5})$ obtained for N={0,3000} is investigated as a function of its transformed co-ordinate $\sqrt{\frac{t\cdot5}{2\pi}}-\left\lfloor\sqrt{\frac{t\cdot5}{2\pi}}\right\rfloor$ value into the rescaled interval [0,1) for each $N \bmod 5$ value. A multi-modal density distribution for the relative positions of the extreme peaks is observed which varies across the four 5-periodic functions. One possible explanation for the relative positions of the modes in the density distribution is presented arising from a heuristic approximation of the indefinite integral of the Riemann Siegel Z function analogues using the tapered finite Dirichlet Series sum. In particular, the indefinite integral approximation of finite Dirichlet Series sum exhibits piecewise behaviour on the critical line with clear mesoscale features that consistently have the steepest curvature in the interval $[N^2\cdot\frac{2\pi}{5},(N+1)^2\cdot\frac{2\pi}{5})$. Of further interest, the indefinite integral approximation exhibits consistent growth of $(\frac{t\cdot 5}{2\pi})^{0.25}$ on the critical line for the investigated t intervals and there are mesoscale differences for the piecewise intervals between three different Dirichlet Series sum approaches to the indefinite integral approximation, the zeroth order Riemann Siegel formula Dirichlet Series sum, and the tapered Dirichlet Series sum results truncated at the first quiescent region ($\lfloor(\frac{t}{\pi})^{(d=1)}\cdot (N_C=5)\rfloor$) and second quiescent region ($\lfloor\sqrt{(\frac{t}{2\pi})^{(d=1)}\cdot (N_C=5)}\rfloor$) respectively for these first degree functions with conductor value $N_C=5$.