This is a basic question concerning the local Langlands correspondence for $GL_2(\mathbb{Q}_p)$, in particular the compatibility with tensor products . I apologize if this is too basic but I hope someone with more experience can answer it quickly.
Let $G=GL_2(\mathbb{Q}_p)$, $B \subset G$ the subgroup of upper triangular matrices and $St_G$ be the Steinberg representation of $G$; this is the unique irreducible quotient of the representation $Ind_B^G(|\cdot|^{1/2},|\cdot|^{-1/2})$ (normalized induction).
According to the local Langlands correspondence for $G$, we can attach to $St_G$ a Weil-Deligne representation $(V=\mathbb{C}^2,\sigma:W_F \rightarrow GL(V),N \in End(V))$: in this case the inertia acts trivially, $\sigma(Fr) = \begin{pmatrix} p^{-1/2} & 0 \\\ 0 & p^{+1/2} \end{pmatrix}$ as a representation of $W_F$ and $N=\begin{pmatrix} 0 & 1 \\\ 0 & 0 \end{pmatrix}$.
Now consider the equality of $L$-factors $L(St_G \times St_G,s)=L(\sigma \otimes \sigma,s)$. According to Jacquet (Automorphic Forms on GL(2), Part II, p. 23), we have $L(St_G \times St_G,s)=(1-q^{-s-1})^{-1}$, an $L$-function of degree $1$. On the RHS, the tensor product of Weil-Deligne representations is defined by $(V,\sigma,N) \otimes (V',\sigma',N')=(V\otimes V',\sigma \otimes \sigma',N \otimes 1 + 1 \otimes N')$. But, if I am not mistaken, $\dim_\mathbb{C} Ker(N \otimes 1 + 1 \otimes N')=2$ so that the RHS seems to be of degree $2$! Where did I go wrong?
