Fundamental properties of symmetric square $L$-functions. I

Sankaranarayanan, A.

doi:10.1215/ijm/1258136138

Cited by 26 publications

(20 citation statements)

References 30 publications

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“…In case of the full modular group, bounds for certain averages of symmetric square L-functions have been proved for example in [10,13,20].…”

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confidence: 99%

On the central value of symmetric square L-functions

Blomer

2008

Math. Z.

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Let S k (N , χ) be the space of cusp forms of weight k, level N and character χ. For f ∈ S k (N , χ) let L(s, sym 2 f ) be the symmetric square L-function and L(s, f ⊗ f ) be the Rankin-Selberg square attached to f . For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym 2 f ) and L(s, f ⊗ f ) over a basis of S k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym 2 f ) does not vanish.

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“…In case of the full modular group, bounds for certain averages of symmetric square L-functions have been proved for example in [10,13,20].…”

mentioning

confidence: 99%

On the central value of symmetric square L-functions

Blomer

2008

Math. Z.

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“…The reason why we can obtain such sharp estimates as above in the Ikeda lift case is surely the existence of decomposition (1.1). The usefulness of such kind of decompositions in mean value problems was first noticed by the second author [16] for Rankin-Selberg L-functions, and then developed in a more general setting by the first author [12]. Among (1.2) -(1.7) proved in [12], the results (1.2), (1.4) and (1.5) determine completely the real order of I (σ ; T ) up to constant factors.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 95%

On the mean square of standard L-functions attached to Ikeda lifts

Matsumoto

Sankaranarayanan

2006

Math. Z.

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By using estimates on the frequency of large values of the Riemann zetafunction and modular L-functions attached to the full modular group SL(2, Z), we prove sharp upper and lower estimates of the mean square of standard L-functions attached to Siegel cusp forms which are Ikeda lifts, on boundaries and the central line of the critical strip. The mean square of spinor L-functions attached to Saito-Kurokawa lifts is also studied.

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“…In view of (1.8), the series for B(s) converges absolutely when Re s > 1, but B(s) has an analytic continuation that is holomorphic when Re s > 0. This important fact follows from Shimura's work [16] (see also Sankaranarayanan [13]), and it implies that (1.7), that is, Z(s) = ζ(s)B(s), holds when Re s > 0 and not only when Re s > 1. The function B(s) is of degree three in S, as its functional equation (see, for instance, Sankaranarayanan [13]) is…”

Section: Introductionmentioning

confidence: 87%