### Abstract

We consider a family of approximations of a Hecke L-function L_{f}(s) attached to a holomorphic cusp form f of positive integral weight k with respect to the full modular group. These families are of the form (Formula presented.), where s= σ+ it is a complex variable and a(n) is a normalized Fourier coefficient of f. From an approximate functional equation, one sees that L_{f}(X; s) is a good approximation to L_{f}(s) when X= t/ 2 π. We obtain vertical strips where most of the zeros of L_{f}(X; s) lie. We study the distribution of zeros of L_{f}(X; s) when X is independent of t. For X= 1 and 2, we prove that all the complex zeros of L_{f}(X; s) lie on the critical line σ= 1 / 2. We also show that as T→ ∞ and X= T^{o(1)}, 100% of the complex zeros of L_{f}(X; s) up to height T lie on the critical line. Here by 100% we mean that the ratio between the number of simple zeros on the critical line and the total number of zeros up to height T approaches 1 as T→ ∞.

Original language | English (US) |
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Pages (from-to) | 391-419 |

Number of pages | 29 |

Journal | Ramanujan Journal |

Volume | 41 |

Issue number | 1-3 |

DOIs | |

State | Published - Nov 1 2016 |

### Keywords

- Approximate functional equation
- Hecke L-functions
- Proportion of zeros on the critical line

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Ramanujan Journal*,

*41*(1-3), 391-419. https://doi.org/10.1007/s11139-016-9791-3