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For Riemann zeta function and $L$-functions of number fields, the width of critical strip is $1$. For $L$-functions of modular forms of weight $k$, the width of the critical strip is $k$.

Why is there a variation in the width of the critical strip for various $L$-functions? Is there a conceptual explanation or an underlying heuristics?

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Roughly speaking the critical strip is where the $L$-function is hard to compute/understand. Its right edge is the boundary of the region where the Dirichlet series/Euler product converges nicely (absolutely, locally uniformly etc). Its left edge is the image of the right edge under the functional equation (to compute it to the left of the critical strip you can use the functional equation to reduce it to comuting a nice series/product).

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