I'd like to ask the following easy question, since I can't find a reference.


Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in terms of $L(H^i(X),s)$. Do they differ by only a shift in $s$?



L-function of twist
The answer is yes. On page 12 of [1], line 7 from the top, it reads
$$ L(M (n), s) = L(M, s + n). $$


The article deals with compatible systems of cohomological realisations (one way to think and work with motives). Every variety gives rise to such a system, and every such system gives you an $L$-function.


Actually, if you carefully trace the definitions, this identity rolls out fairly easy. For a good overview of the definitions, see [2].




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[1]: Minhyong Kim. *“An introduction to motives I: classical motives and motivic $L$-functions”* (available at <http://ucl.ac.uk/~ucahmki/ihes3.pdf> ).


[2]: Serre, Jean-Pierre. *“Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures).”* (available at <http://archive.numdam.org/article/SDPP_1969-1970__11_2_A4_0.pdf>)


