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`\DeclareMathOperator`; grammar
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LSpice
LSpice
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Whether or not $\DeclareMathOperator\sym{sym}$Does $L(s, \text{sym}^2\sym^2 f \times \text{sym}^2\sym^2 g)$ hashave a pole at $s=1$?

I encounter$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the polepoles of $GL_3\times GL3$$\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts here.

Let $f$ be aan $SL_2$$\SL_2$ newform of level $P$ (such as a primitive holomorphic form) and $g$ a newform with trivial level. Does any expert here know that whether or not the $L$-function $L(s, \text{sym}^2 f \times \text{sym}^2 g)$$L(s, \sym^2 f \times \sym^2 g)$ has a pole at $s=1$? If not, how about the upper-bound of the $L$-value at $s=1$? I guess $L(1, \text{sym}^2 f \times \text{sym}^2 g)\ll P^\varepsilon$$L(1, \sym^2 f \times \sym^2 g)\ll P^\varepsilon$, but I don't know any ideas of the proof.

If any experts here leanknow something onabout this question, please give a guide sharing some of your valuable comments or give some references. Thanks in advance and thanks for your time.

Whether or not $L(s, \text{sym}^2 f \times \text{sym}^2 g)$ has a pole at $s=1$?

I encounter a question on the pole of $GL_3\times GL3$ $L$-function, which needs the knowledge of the experts here.

Let $f$ be a $SL_2$ newform of level $P$ (such as a primitive holomorphic form) and $g$ a newform with trivial level. Does any expert here know that whether or not the $L$-function $L(s, \text{sym}^2 f \times \text{sym}^2 g)$ has a pole at $s=1$? If not, how about the upper-bound of the $L$-value at $s=1$? I guess $L(1, \text{sym}^2 f \times \text{sym}^2 g)\ll P^\varepsilon$, but I don't know any ideas of the proof.

If any experts here lean something on this question, please give a guide sharing some of your valuable comments or give some references. Thanks in advance and thanks for your time.

$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts here.

Let $f$ be an $\SL_2$ newform of level $P$ (such as a primitive holomorphic form) and $g$ a newform with trivial level. Does any expert here know whether or not the $L$-function $L(s, \sym^2 f \times \sym^2 g)$ has a pole at $s=1$? If not, how about the upper-bound of the $L$-value at $s=1$? I guess $L(1, \sym^2 f \times \sym^2 g)\ll P^\varepsilon$, but I don't know any ideas of the proof.

If any experts here know something about this question, please give a guide sharing some of your valuable comments or give some references.

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GH from MO
GH from MO
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guys

I encounter a question on the pole of $GL_3\times GL3$ $L$-function, which needs the knowledge of the experts here.

Let $f$ be a $SL_2$ newform of level $P$ (such as a primitive holomorphic form) and $g$ a newform with trivial level. Does any expert here know that whether or not the $L$-function $L(s, \text{sym}^2 f \times \text{sym}^2 g)$ has a pole at $s=1$? If not, how about the upper-bound of the $L$-value at $s=1$? I guess $L(1, \text{sym}^2 f \times \text{sym}^2 g)\ll P^\varepsilon$, but I don't know any ideas of the proof.

If any experts here lean something on this question, please give a guide sharing some of your valuable comments or give some references. Thanks in advance and thanks for your time.

guys

I encounter a question on the pole of $GL_3\times GL3$ $L$-function, which needs the knowledge of the experts here.

Let $f$ be a $SL_2$ newform of level $P$ (such as a primitive holomorphic form) and $g$ a newform with trivial level. Does any expert here know that whether or not the $L$-function $L(s, \text{sym}^2 f \times \text{sym}^2 g)$ has a pole at $s=1$? If not, how about the upper-bound of the $L$-value at $s=1$? I guess $L(1, \text{sym}^2 f \times \text{sym}^2 g)\ll P^\varepsilon$, but I don't know any ideas of the proof.

If any experts here lean something on this question, please give a guide sharing some of your valuable comments or give some references. Thanks in advance and thanks for your time.

I encounter a question on the pole of $GL_3\times GL3$ $L$-function, which needs the knowledge of the experts here.

Let $f$ be a $SL_2$ newform of level $P$ (such as a primitive holomorphic form) and $g$ a newform with trivial level. Does any expert here know that whether or not the $L$-function $L(s, \text{sym}^2 f \times \text{sym}^2 g)$ has a pole at $s=1$? If not, how about the upper-bound of the $L$-value at $s=1$? I guess $L(1, \text{sym}^2 f \times \text{sym}^2 g)\ll P^\varepsilon$, but I don't know any ideas of the proof.

If any experts here lean something on this question, please give a guide sharing some of your valuable comments or give some references. Thanks in advance and thanks for your time.

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hofnumber
hofnumber
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Whether or not $L(s, \text{sym}^2 f \times \text{sym}^2 g)$ has a pole at $s=1$?

guys

I encounter a question on the pole of $GL_3\times GL3$ $L$-function, which needs the knowledge of the experts here.

Let $f$ be a $SL_2$ newform of level $P$ (such as a primitive holomorphic form) and $g$ a newform with trivial level. Does any expert here know that whether or not the $L$-function $L(s, \text{sym}^2 f \times \text{sym}^2 g)$ has a pole at $s=1$? If not, how about the upper-bound of the $L$-value at $s=1$? I guess $L(1, \text{sym}^2 f \times \text{sym}^2 g)\ll P^\varepsilon$, but I don't know any ideas of the proof.

If any experts here lean something on this question, please give a guide sharing some of your valuable comments or give some references. Thanks in advance and thanks for your time.