# Clarification in the definition of Critical strip for L functions (in particular of modular cusp forms and may be zeta function )

Let $$f$$ be a cusp form of weight $$k$$ with respect to $$SL_2(\mathbb{Z})$$. Define its associated L-series, $$L(f,s)$$ by $$\sum_{n=1}^{\infty} \frac{a_f(n)}{n^s}$$. One knows by Ramanujan-Petersson theorem(Deligne) that, $$L(f,s)$$ is holomorphic on $$\left\{Re(s)>\frac{k+1}{2}\right\}$$. Also, we know that the analytic continuation(Mellin Transform of $$f(it)$$) of the above L-series, $$\Lambda(f,s)(=(2\pi)^{-s}\Gamma(s)L(f,s))$$ satisfies a functional equation relating s to $$k-s$$. Now comes my possibly silly questions/clarifications, all in the setting of elliptic modular forms for integral weight $$k$$ and $$SL_2(\mathbb{Z})$$.

1)Whenever the term "critical strip" comes in the conversation in literature (Books and Papers), people mostly speak of eigen-functions and there is always a normalisation automatically tagging alongside. As I understand the critical strip, it is that region where we don't have absolute (and uniform) convergence of the original L series and where we need some non-trivial study of $$\Lambda$$. (By the symmetry of the functional equation of $$\Lambda$$, anyway the region to the left of the $$Re(s)= \frac{k-1}{2}$$ is automatically handled). Thus, I can still technically speak of the critical strip for an L-function associated to any cusp form too(not necessarily normalised eigen forms) right? Why it seems everybody is interested $$\textbf{only}$$ on normalised eigen forms? I do understand that values of $$L(f,s)$$ are non-zero outside the critical strip if $$a_f(n)$$-s are multiplicative due to the existence of the Infinite Product which is why studying Eigen forms make sense. But do number theorists study about zeros and non vanishing (both inside and outside the critical strip) of L(f,s) for general cusp forms(Q1a) too? (any reference/info would be appreciated!)

2a)The other questions are related to the length of the critical strip for L functions of eigen forms, normalised (algebraic normalisation as mentioned in Wikipedia).https://en.wikipedia.org/wiki/Eigenform. From the Deligne bound and the functional equation of $$\Lambda$$, the strip seems to be between $$\frac{k-1}{2}$$ and $$\frac{k+1}{2}$$ and hence of length $$1$$. If $$f$$ is just a general cusp form, is the length still $$1$$?

2b) If $$f \in S_k$$, when are its fourier coefficients algebraic? What is the weight of L(f,s) in this case? (Definition as per LMFDB) In that case, is this algebraic normalisation the same as the arithmetic normalisation? (arithmetic normalisation as per LMFDB https://www.lmfdb.org/knowledge/show/lfunction.normalization)?

3a)What if I consider non-normalised cusp form $$f\in S_k(1)$$? Will the length of critical strip be changed? Will the length be $$a_f(1)$$ in that case?

3b) Zagier, Kohnen in their Inventiones paper 1981, "Values of L-series of modular forms at the center of the critical strip", writes as below:

"$$\dots$$ and that the values of L(f,s) at integral arguments within the$$\textit{critical strip 0< Re(s)<2k}$$ can be expressed as algebraic multiples of certain "periods" associated to f$$\dots$$".

Here they are considering only cusp eigen forms(weight 2k). This seems to be contradicting the claim in (2) and also seem to contradict LMFDB definition of critical strip(https://www.lmfdb.org/knowledge/show/lfunction.critical_strip)!

However, I also saw an query (Why is width of critical strip what it is?) which says that the length of the critical strip for a modular form of weight $$k$$ is $$k$$. Where am I wrong then? Refer 3b too.

4) Are the definitions of Analytic normalisation in wikipedia and LMFDB consistent? Analytic normalisation (Wikipedia vs LMFDB) https://en.wikipedia.org/wiki/Eigenform and https://www.lmfdb.org/knowledge/show/lfunction.normalization. Is Arithmetic normalisation(LMFDB) = Algebraic normalisation(Wikipedia)?

• For a level $$1$$ cusp form $$f\in S_k(SL_2(\Bbb{Z}))$$ then $$(2\pi)^{-s} \Gamma(s)\sum_n a_n(f) n^{-s}=\int_0^\infty f(iy)y^{s-1}dy$$ is entire (the integral converges everywhere, not the Dirichlet series) and invariant under $$s\to k-s$$. If it is an eigenform then it has an Euler product $$\sum_n a_n(f) n^{-s}=\prod_p \frac1{1-a_p(f)p^{-s}+p^{2k-1-2s}}$$.

• To show that up to a shift it is a Selberg class L-function (here I'll assume $$f\in S_k(\Gamma_1(N))$$ is a newform of any level) you need the Ramanujan Deligne $$|a_p(f)|\le C p^{(k-1)/2}$$ theorem you mention (the elementary bound is $$\le C p^{k/2}$$) plus the newform theorems $$\sum_n a_nn^{-s}= \prod_p \frac1{1-a_p(f) p^{-s-(k-1)/2}+\chi(p) p^{-2s}}$$ with $$\chi$$ a Dirichlet character $$\bmod N$$ and $$f(-1/(Nz)) = \varepsilon f^*(z)$$ where $$f^*\in S_k(\Gamma_1(N))$$ is the newform with complex conjugate coefficients, from there both $$L(s,f) = \sum_n a_n n^{-s-(k-1)/2}$$ and its Euler product converge for $$\Re(s) > 1$$ , the functional equation is $$\Lambda(s,f)=N^{s/2}(2\pi)\Gamma(s)L(s,f)=\varepsilon\Lambda(1-s,f^*)$$, then quite all the theorems about $$\zeta(s),L(s,\chi)$$ stay true for $$L(s,f)$$, in particular the critical strip is $$\Re(s)\in [0,1]$$ and $$L(s,f)$$ has a Riemann hypothesis stating that all its non-trivial zeros are on $$\Re(s)=1/2$$ ie. that they are zero-crossings of the real function $$\varepsilon^{-1/2}\Lambda(1/2+it,f)$$.

• The normalization is just to ensure $$f$$ is a newform, $$L(s,f)$$ is in the Selberg class and the coefficients are also the eigenvalues of the Hecke operators. Changing finitely many Euler factors of a Selberg class L-function adds some Euler factor into the functional equation and some trivial zeros distributed on vertical lines.

For any cusp form $$\int_0^\infty g(iy)y^{s-1}dy$$ thus $$\Gamma(s)L(s,g)$$ is entire.

The oldform theorem is that any cusp form $$g\in S_k(\Gamma_1(N))$$ is of the form $$\sum_j c_j f_j(d_j z)$$ for some newforms $$f_j\in S_k(\Gamma_1(N/d_j))$$.

If $$g$$ is an eigenform only for the Hecke operators $$T_n,n\nmid N$$ then find the newform $$f\in S_k(\Gamma_1(N/m))$$ underlying $$g$$ you'll obtain $$g(z) = \sum_{d | m} c_d f(dz)$$.

The L-functions of the Eisenstein series are linear combinations of $$L(s+(k-1)/2,\chi_1)L(s-(k-1)/2,\chi_2)$$, the new-Eisenstein are those with $$\chi_1,\chi_2$$ primitive characters $$\bmod a,b$$ (from the functional equation I see the level is $$ab$$)

The Riemann hypothesis for linear combinations of L-functions is to replace statements about the asymptotic of $$\sum_{p\le k} a_p(f) p ^{-s}$$ by statements about the asymptotic of $$\sum_{\gcd(n,\prod_{p\le k})=1} a_n(f)n^{-s}$$. Non-trivial linear combinations have infinitely many zeros on $$\Re(s)\in (1,1+\epsilon)$$, this is because that the $$\log p$$ are $$\Bbb{Q}$$-linearly independent imply the values of an Euler product (converging absolutely for $$\Re(s) > 1$$ but not at $$s=1$$) are dense in $$\Bbb{C}^*$$

• If $f$ is just a modular form in $S_k(1)$, how is its L function having width(critical strip) $k$? Commented Jan 2, 2020 at 11:02
• It doesn't. But $f(-1/z) = z^k f(z)$ means $(2\pi)^{-s} \Gamma(s)\sum_n a_n(f) n^{-s}=\int_0^\infty f(iy)y^{s-1}dy$ is invariant under $s\to k-s$ Commented Jan 2, 2020 at 11:18
• Can we take a simple case for my clarity? Say $f$ is in $S_k(1)$. Due to holomorphicity, L series at $s=\frac{k+1+\epsilon}{2}$ is finite. Under the invariance "s goes to k-s", once I know that value at $\frac{k+1+\epsilon}{2}$, I also know the value at $\frac{k-1-\epsilon}{2}$. But Zagier is saying critical strip between $0$ and $k$. Shouldn't it be between $\frac{k-1}{2}$ and $\frac{k+1}{2}$? This is still confusing me. Please explain in a basic way. I am new to this subject. Commented Jan 2, 2020 at 11:51
• Yes, Zagier is wrong if he is saying that, $\sum_n a_n(f) n^{-s}$ is not more mysterious on $((k+1)/2,k)$ than $\zeta(s)$ is on $(1,(k+1)/2)$. Something special arises with the Eisenstein series : this time $L(s,E_k)$ being $\zeta(s)\zeta(s-k)$ there are 2 critical strips, still of width $1$ and centered at $1/2,k-1/2$ Commented Jan 2, 2020 at 14:51