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Let $f(z)=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}e(nz)\in S_{k+1/2}(\Gamma_0(4N))$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{|a(p)|^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\frac{|a(n)|^2}{n^s}$$ Is there a functional equation link the two series ?

Let $f(z)=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}e(nz)\in S_{k+1/2}(\Gamma_0(4N),\chi)$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{|a(p)|^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\frac{|a(n)|^2}{n^s}$$ Is there a functional equation link the two series ?

Let $f=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}\in S_{k+1/2}(\Gamma_0(4N))$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{a(p)^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\frac{a(n)^2}{n^s}$$ Is there a functional equation link the two series ?

Let $f(z)=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}e(nz)\in S_{k+1/2}(\Gamma_0(4N))$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{|a(p)|^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\frac{|a(n)|^2}{n^s}$$ Is there a functional equation link the two series ?

Let $f=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}\in S_{k+1/2}(\Gamma_0(4))$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{a(p)^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\frac{a(n)^2}{n^s}$$ Is there a functional equation link the two series ?

Let $f=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}\in S_{k+1/2}(\Gamma_0(4N))$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{a(p)^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\frac{a(n)^2}{n^s}$$ Is there a functional equation link the two series ?