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36 votes
1 answer
4k views

Special values of L-functions as periods

If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles. For example, when $M=\...
Joël's user avatar
  • 26k
11 votes
1 answer
698 views

Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?

Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion $$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$ and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...
D_S's user avatar
  • 6,180
7 votes
3 answers
826 views

Analytic equivalents for primes in arithmetic progressions

By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$. I would like ...
Greg Martin's user avatar
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18 votes
1 answer
562 views

Is special value of Epstein zeta function in 3 variables a period?

Kontsevich-Zagier's article "Periods" contains the following question Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period? ($\sum'$ means we do not sum ...
Pig's user avatar
  • 809
5 votes
4 answers
4k views

Values of Dirichlet L-funcions at natural numbers

I want to know about reference of formulas for $$ L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s} $$ for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have ...
emiliocba's user avatar
  • 2,446
5 votes
2 answers
941 views

$B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$

Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression $$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho ...
H A Helfgott's user avatar
  • 20.2k