Let $f_1$, $f_2$ be two fixed distinct holomorphic cuspidal Hecke eigenforms of level 1 and respective weights ${\kappa }_1$, ${\kappa }_2$ with ${\kappa }_1\equiv {\kappa }_2(\mathrm{mod}4)$. Let $L(s,f_1\otimes \chi )$, $L(s,f_2\otimes \chi )$ denote the corresponding twisted L-functions associated to $f_1$, $f_2$ twisted by a primitive Dirichlet character χ modulo q. In this paper, we obtain a lower bound for the 2kth moment of the product of two twisted L-functions at the central value for all sufficiently factorable q including 99.9% of all admissible moduli q, i.e., for any natural number k,$$\begin{gathered} \underset{\chi \mathrm{mod}q}{{\sum }^{⁎}}{(L(1/2,f_1\otimes \chi )\overline{L(1/2,f_2\otimes \chi )})}^{2k} \\ {\gg }_k\psi (q){(\log q)}^{2k^2}{(\log \log q)}^{-4k^2}, \end{gathered}$$ where $\psi (q)$ denotes the number of primitive characters modulo q and the asterisk restricts the summation to all primitive characters, which coincides with the remark of Blomer and Milićević [1].

让 $F_{\mathrm{1个}}$， $F_2$ 是1级和相应权重的两个固定的不同全同形尖齿Hecke本征形 ${\kappa }_{\mathrm{1个}}$， ${\kappa }_2$ 与 ${\kappa }_{\mathrm{1个}}\equiv {\kappa }_2（\mathrm{模}4）$。让$\mathrm{大号}（s，F_{\mathrm{1个}}\otimes \chi ）$， $\mathrm{大号}（s，F_2\otimes \chi ）$表示与之相关的相应的扭曲L函数$F_{\mathrm{1个}}$， $F_2$由原始Dirichlet字符χ模q扭曲。在本文中，对于所有充分分解的q，包括所有容许模数q的99.9％，即对于任何自然数k，我们在中心值处获得了两个扭曲L函数的乘积的第2 k矩的下界。$$\begin{gathered} \underset{\chi \mathrm{模}q}{{\sum }^{⁎}}{（\mathrm{大号}（\mathrm{1个}/2，F_{\mathrm{1个}}\otimes \chi ）\stackrel{〜}{\mathrm{大号}（\mathrm{1个}/2，F_2\otimes \chi ）}）}^{2ķ} \\ {\gg }_{ķ}\psi （q）{（\mathrm{日志}q）}^{2{ķ}^2}{（\mathrm{日志}\mathrm{日志}q）}^{-4{ķ}^2}， \end{gathered}$$ 哪里 $\psi （q）$表示以q为模的原始字符的数量，星号将总和限制为所有原始字符，这与Blomer和Milićević[1]的说法相吻合。