Functiones et Approximatio Commentarii Mathematici

Le premier coefficient négatif des fonctions $L$ de puissances symétriques

Kamel Mazhouda, Khadija Mbarki, and Jie Wu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Désignons par $\lambda_{{\rm sym}^mf}(n)$ le $n$-ème coefficient dans la série de Dirichlet représentant la fonction $L$ de puissances symétriques $L(s, {\rm sym}^mf)$ associée à une forme primitive $f$ de poids $k$ et de niveau $N$. Dans ce papier, on étudie la taille de l'entier le plus petit $n$ tel que $\lambda_{{\rm sym}^mf}(n)<0$ et $(n,N)=1$. En désignant par $n_{{\rm sym}^mf}$ cet entier, on montre que $$ n_{{\rm sym}^3f} \ll (k^{4} N^3)^{6/31} \qquad\text{et}\qquad n_{{\rm sym}^4f} \ll (k^{4} N^4)^{5/36}, $$ où les constantes impliquées sont absolues.

Article information

Funct. Approx. Comment. Math., Volume 56, Number 2 (2017), 239-258.

First available in Project Euclid: 28 March 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F12: Automorphic forms, one variable 11F30: Fourier coefficients of automorphic forms
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Formes modulaires fonctions $L$ coefficients de Fourier signes


Mazhouda, Kamel; Mbarki, Khadija; Wu, Jie. Le premier coefficient négatif des fonctions $L$ de puissances symétriques. Funct. Approx. Comment. Math. 56 (2017), no. 2, 239--258. doi:10.7169/facm/1609.

Export citation


  • D.R. Heath-Brown, Convexity bounds for $L$-functions, Acta Arith. 136 (2009), no. 4, 391–395.
  • J. Cogdell and P. Michel, On the complex moments of symmetric power $L$-functions at $s=1$, Int. Math. Res. Not. 31 (2004), 1562–1618.
  • P. Deligne, La conjecture de Weil, I, Publ. Math. IHES 48 (1974), 273–308.
  • P. Deligne, La conjecture de Weil, II, Publ. Math. IHES 52 (1981), 313–428.
  • G. Gelbart, Notes on Langlands pictures of Automorphic forms and $L-$functions. Lecture $X$: Langlands program, (Mars 2009).
  • S. Gelbart and H. Jacquet, A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542.
  • H.H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197.
  • W. Kohnen, J. Sengupta and H. Iwaniec, The first negative Hecke eigenvalue, Int. J. Number Theory 3 (2007), 355–363.
  • E. Kowalski, Y.-K. Lau, K. Soundararajan and J. Wu, On modular signs, Math. Proc. Camb. Phil. Soc. 149 (2010), 389–411.
  • Y.-K. Lau, J.-Y. Liu and J. Wu, The first negative coefficient of symmetric square $L$-functions, Ramanujan J. 27 (2012), 419–441.
  • J.-Y. Liu, Y. Qu and J. Wu, Two Linnik-type problems for automorphic $L$-functions, Math. Proc. Camb. Phil. Soc. 151 (2011), 219–227.
  • K. Matomäki, On signs of Fourier coefficients of cusp forms, Math. Proc. Cambridge. Philos. Soc. 152 (2012), 207–222.
  • Y. Qu, Linnik-type problems for automorphic $L$-functions, J. Number Theory 130 (2010), no. 3, 786–802.
  • E. Royer and J. Wu, Special values of symmetric power $L$-functions and Hecke eigenvalues, Journal de théorie des nombres de bordeaux 19 (2007), 703–753.