This is of course a very-well known problem, but still let me ask the questions my way. Let $L(s)$ be a "motivic" $L$-function, whatever that means: in particular, it has an Euler product (including at "bad" primes), and a (possibly conjectural) functional equation of standard type with an arithmetic conductor $N$ occurring as $N^{s/2}$ in the factor at infinity. Let $p$ be a prime, denote by $L\_p(s)$ the denominator of the Euler factor at $p$, by $d\_p$ the degree of $L\_p(s)$ in $p^{-s}$, and by $v\_p(N)$ the valuation of $N$ at $p$. Finally let $d$ be the degree of the $L$-function, equal both to the number of $\Gamma\_{\mathbb R}(s)$ factors and to the generic degree in $p^{-s}$ of $L\_p(s)$. 


I define a prime to be good if it does not divide $N$ (note that if $L$ comes from a variety with bad reduction at $p$, it may still be a good prime in this sense). For the following statements, please tell me if they are true, or only sometimes true, sometimes false (and in that case possibly some extra info):


1) If $p$ is good then $d\_p=d$.


2) If $d\_p=d$ then $p$ is good.


3) $v\_p\ge d-d\_p$


4) I know that one can have $v\_p>d-d\_p$, but can one give a reasonably simple criterion for equality ?



Euler factors of L-function at bad primes
1) True.


2) True.


3) True.


4) They are equal in the case of "tame ramification", e.g. if the degree of $K(\mu\_p)$ over $K$, $K$ the coefficient field of the motive, is greater than $d$. 


One just has to recall that for $V$ the $\ell$-adic Galois representation associated to the motive, $\dim V$ is the degree of the $L$-function, the local $L$-factor is the characteristic polynomial of $\operatorname{Frob}\_p$ on the invariants of the inertia group at $p$, and a prime is good if and only if the inertia group acts nontrivially. Furthermore $v\_p$ is the Artin conductor which consists of the codimension of the inertia invariant subspace (i.e. $d-d\_p$) plus the Swan conductor.


The Swan conductor vanishes if and only if the wild inertia group acts trivially. The last claim depends on something that I think is still conjectural, depending on the theory of motives, which is that the characteristic polynomials of elements of the Weil group are defined over $K$. From this, it follows immediately, as these characteristic polynomials have degree $d$, and for elements of teh wild inertia group, roots $p$-power roots of unity.


