Functional equation for $GL(3)times GL(2)times GL(1)$ L-functions
$begingroup$
For two Maass forms
$$f(z)=sum_{nneq 0}a(n)sqrt{2pi y}K_{v_1-frac{1}{2}}(2pi|n|y)e^{2pi inx}$$ and$$g(z)=sum_{gammain U_2(mathbb{Z})backslash SL(2,mathbb{Z})} ,,,,,sum_{m=1}^{infty},,sum_{nneq 0}frac{b(m,n)}{|mn|}W_{text{Jacquet}}left(begin{pmatrix} |mn| & & \
& m & \
& & 1 end{pmatrix}begin{pmatrix}gamma & \ & 1end{pmatrix}z,,, v_2,,psi_{1,frac{n}{|n|}} right)$$ for $SL(2,mathbb{Z}), SL(3,mathbb{Z})$, respectively (assume $f$ is to be even), we know the functional equation of the Rankin-Selberg $L$-function
$$L_{ftimes g}(s)=sum_{m=1}^{infty}sum_{n=1}^{infty}frac{a(n)b(m,n)}{(m^2n)^s} $$ where $a(n),b(m,n)$ are the coefficients in the Fourier-Whittaker expansions of $f$ and $g$ resp. This follows from the following integral representation
$$int_{SL(2,mathbb{Z})backslashmathbb{H}}f(z).gleft(begin{pmatrix}z&\&1 end{pmatrix}right)|det(z)|^{s-frac{1}{2}}d^{*}z=L_{ftimes g}(s)G_{(v_1,v_2)} $$ Do we know functional equation for the twisted series
$$L_{f_{chi}times g}(s)=sum_{m=1}^{infty}sum_{n=1}^{infty}frac{chi(n)a(n)b(m,n)}{(m^2n)^s} $$ where $chi$ is say primitive mod $p$(prime) ?
analytic-number-theory modular-forms automorphic-forms langlands-program
asked Dec 23 '18 at 14:56
pkspks
18411
$endgroup$
add a comment |
$begingroup$
For two Maass forms
$$f(z)=sum_{nneq 0}a(n)sqrt{2pi y}K_{v_1-frac{1}{2}}(2pi|n|y)e^{2pi inx}$$ and$$g(z)=sum_{gammain U_2(mathbb{Z})backslash SL(2,mathbb{Z})} ,,,,,sum_{m=1}^{infty},,sum_{nneq 0}frac{b(m,n)}{|mn|}W_{text{Jacquet}}left(begin{pmatrix} |mn| & & \
& m & \
& & 1 end{pmatrix}begin{pmatrix}gamma & \ & 1end{pmatrix}z,,, v_2,,psi_{1,frac{n}{|n|}} right)$$ for $SL(2,mathbb{Z}), SL(3,mathbb{Z})$, respectively (assume $f$ is to be even), we know the functional equation of the Rankin-Selberg $L$-function
$$L_{ftimes g}(s)=sum_{m=1}^{infty}sum_{n=1}^{infty}frac{a(n)b(m,n)}{(m^2n)^s} $$ where $a(n),b(m,n)$ are the coefficients in the Fourier-Whittaker expansions of $f$ and $g$ resp. This follows from the following integral representation
$$int_{SL(2,mathbb{Z})backslashmathbb{H}}f(z).gleft(begin{pmatrix}z&\&1 end{pmatrix}right)|det(z)|^{s-frac{1}{2}}d^{*}z=L_{ftimes g}(s)G_{(v_1,v_2)} $$ Do we know functional equation for the twisted series
$$L_{f_{chi}times g}(s)=sum_{m=1}^{infty}sum_{n=1}^{infty}frac{chi(n)a(n)b(m,n)}{(m^2n)^s} $$ where $chi$ is say primitive mod $p$(prime) ?
analytic-number-theory modular-forms automorphic-forms langlands-program
asked Dec 23 '18 at 14:56
pkspks
18411
$endgroup$
$begingroup$
Won't it work the same way with $int_{Gamma_1(p)setminus H} f_chi(z) (...) $ ?
$endgroup$
– reuns
Dec 25 '18 at 21:52
$begingroup$
after doing the usual unfolding you will end with integrands of type $sum_{gammainGamma_1(p)backslash Gamma}f_{chi}(gamma z)$. Do we know the Fourier coefficients of this sum is terms of the Fourier coefficients of $f$ ?
$endgroup$
– pks
Dec 26 '18 at 4:11
add a comment |
$begingroup$
For two Maass forms
$$f(z)=sum_{nneq 0}a(n)sqrt{2pi y}K_{v_1-frac{1}{2}}(2pi|n|y)e^{2pi inx}$$ and$$g(z)=sum_{gammain U_2(mathbb{Z})backslash SL(2,mathbb{Z})} ,,,,,sum_{m=1}^{infty},,sum_{nneq 0}frac{b(m,n)}{|mn|}W_{text{Jacquet}}left(begin{pmatrix} |mn| & & \
& m & \
& & 1 end{pmatrix}begin{pmatrix}gamma & \ & 1end{pmatrix}z,,, v_2,,psi_{1,frac{n}{|n|}} right)$$ for $SL(2,mathbb{Z}), SL(3,mathbb{Z})$, respectively (assume $f$ is to be even), we know the functional equation of the Rankin-Selberg $L$-function
$$L_{ftimes g}(s)=sum_{m=1}^{infty}sum_{n=1}^{infty}frac{a(n)b(m,n)}{(m^2n)^s} $$ where $a(n),b(m,n)$ are the coefficients in the Fourier-Whittaker expansions of $f$ and $g$ resp. This follows from the following integral representation
$$int_{SL(2,mathbb{Z})backslashmathbb{H}}f(z).gleft(begin{pmatrix}z&\&1 end{pmatrix}right)|det(z)|^{s-frac{1}{2}}d^{*}z=L_{ftimes g}(s)G_{(v_1,v_2)} $$ Do we know functional equation for the twisted series
$$L_{f_{chi}times g}(s)=sum_{m=1}^{infty}sum_{n=1}^{infty}frac{chi(n)a(n)b(m,n)}{(m^2n)^s} $$ where $chi$ is say primitive mod $p$(prime) ?
analytic-number-theory modular-forms automorphic-forms langlands-program
asked Dec 23 '18 at 14:56
pkspks
18411
$endgroup$
For two Maass forms
$$f(z)=sum_{nneq 0}a(n)sqrt{2pi y}K_{v_1-frac{1}{2}}(2pi|n|y)e^{2pi inx}$$ and$$g(z)=sum_{gammain U_2(mathbb{Z})backslash SL(2,mathbb{Z})} ,,,,,sum_{m=1}^{infty},,sum_{nneq 0}frac{b(m,n)}{|mn|}W_{text{Jacquet}}left(begin{pmatrix} |mn| & & \
& m & \
& & 1 end{pmatrix}begin{pmatrix}gamma & \ & 1end{pmatrix}z,,, v_2,,psi_{1,frac{n}{|n|}} right)$$ for $SL(2,mathbb{Z}), SL(3,mathbb{Z})$, respectively (assume $f$ is to be even), we know the functional equation of the Rankin-Selberg $L$-function
$$L_{ftimes g}(s)=sum_{m=1}^{infty}sum_{n=1}^{infty}frac{a(n)b(m,n)}{(m^2n)^s} $$ where $a(n),b(m,n)$ are the coefficients in the Fourier-Whittaker expansions of $f$ and $g$ resp. This follows from the following integral representation
$$int_{SL(2,mathbb{Z})backslashmathbb{H}}f(z).gleft(begin{pmatrix}z&\&1 end{pmatrix}right)|det(z)|^{s-frac{1}{2}}d^{*}z=L_{ftimes g}(s)G_{(v_1,v_2)} $$ Do we know functional equation for the twisted series
$$L_{f_{chi}times g}(s)=sum_{m=1}^{infty}sum_{n=1}^{infty}frac{chi(n)a(n)b(m,n)}{(m^2n)^s} $$ where $chi$ is say primitive mod $p$(prime) ?
analytic-number-theory modular-forms automorphic-forms langlands-program
analytic-number-theory modular-forms automorphic-forms langlands-program
asked Dec 23 '18 at 14:56
pkspks
18411
asked Dec 23 '18 at 14:56
pkspks
18411
edited Dec 24 '18 at 17:07
pks
asked Dec 23 '18 at 14:56
pkspks
18411
asked Dec 23 '18 at 14:56
pkspks
18411
asked Dec 23 '18 at 14:56
pkspks
18411
18411
$begingroup$
Won't it work the same way with $int_{Gamma_1(p)setminus H} f_chi(z) (...) $ ?
$endgroup$
– reuns
Dec 25 '18 at 21:52
$begingroup$
after doing the usual unfolding you will end with integrands of type $sum_{gammainGamma_1(p)backslash Gamma}f_{chi}(gamma z)$. Do we know the Fourier coefficients of this sum is terms of the Fourier coefficients of $f$ ?
$endgroup$
– pks
Dec 26 '18 at 4:11
add a comment |
$begingroup$
Won't it work the same way with $int_{Gamma_1(p)setminus H} f_chi(z) (...) $ ?
$endgroup$
– reuns
Dec 25 '18 at 21:52
$begingroup$
after doing the usual unfolding you will end with integrands of type $sum_{gammainGamma_1(p)backslash Gamma}f_{chi}(gamma z)$. Do we know the Fourier coefficients of this sum is terms of the Fourier coefficients of $f$ ?
$endgroup$
– pks
Dec 26 '18 at 4:11
$begingroup$
Won't it work the same way with $int_{Gamma_1(p)setminus H} f_chi(z) (...) $ ?
$endgroup$
– reuns
Dec 25 '18 at 21:52
$begingroup$
Won't it work the same way with $int_{Gamma_1(p)setminus H} f_chi(z) (...) $ ?
$endgroup$
– reuns
Dec 25 '18 at 21:52
$begingroup$
after doing the usual unfolding you will end with integrands of type $sum_{gammainGamma_1(p)backslash Gamma}f_{chi}(gamma z)$. Do we know the Fourier coefficients of this sum is terms of the Fourier coefficients of $f$ ?
$endgroup$
– pks
Dec 26 '18 at 4:11
$begingroup$
after doing the usual unfolding you will end with integrands of type $sum_{gammainGamma_1(p)backslash Gamma}f_{chi}(gamma z)$. Do we know the Fourier coefficients of this sum is terms of the Fourier coefficients of $f$ ?
$endgroup$
– pks
Dec 26 '18 at 4:11
add a comment |
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$begingroup$
Won't it work the same way with $int_{Gamma_1(p)setminus H} f_chi(z) (...) $ ?
$endgroup$
– reuns
Dec 25 '18 at 21:52
$begingroup$
after doing the usual unfolding you will end with integrands of type $sum_{gammainGamma_1(p)backslash Gamma}f_{chi}(gamma z)$. Do we know the Fourier coefficients of this sum is terms of the Fourier coefficients of $f$ ?
$endgroup$
– pks
Dec 26 '18 at 4:11