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Miller showed that, assuming the generalized Riemann Hypothesis, every entire $L$-function of real archimedian type has a zero in the interval $\\frac12+i t$ with $-t_0 < t < t_0$, where $t_0\\approx 14.13$ corresponds to the first zero of the Riemann zeta function. We give an example of a self-dual degree-4 $L$-function whose first positive imaginary zero is at $t_1\\approx 14.496$. In particular, Miller's result does not hold for general $L$-functions. We show that all $L$-functions satisfying some additional (conjecturally true) conditions have a zero in the interval $(-t_2,t_2)$ wi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.5996","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-26T15:47:45Z","cross_cats_sorted":[],"title_canon_sha256":"2dddd2a20851cbad1ff592883830647833485aec5d1a710531ca266d7cf88454","abstract_canon_sha256":"87688d6fe62c2496ec45e5a16fefe8305b39e6c6711cc4246d248bc8e97527d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:52.752811Z","signature_b64":"PnjUj75zdCTyhOIFQ9kYAAXnXrl1e1eN/2nugjytliQz1+rUjvhvR2TrGS6rZ7hQ0kCUlLZfW28g6wpBA1DUCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"351db7428fb959f92b65981647b913863165aadce2b5c9478395087a6b89efd9","last_reissued_at":"2026-05-18T02:42:52.752398Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:52.752398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The highest lowest zero of general L-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Akio Fujii, David W. Farmer, Hiroyuki Yoshida, J. Brian Conrey, Jonathan Bober, Michael Rubinstein, Sally Koutsoliotas, Stefan Lemurell","submitted_at":"2012-11-26T15:47:45Z","abstract_excerpt":"Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire $L$-function of real archimedian type has a zero in the interval $\\frac12+i t$ with $-t_0 < t < t_0$, where $t_0\\approx 14.13$ corresponds to the first zero of the Riemann zeta function. We give an example of a self-dual degree-4 $L$-function whose first positive imaginary zero is at $t_1\\approx 14.496$. In particular, Miller's result does not hold for general $L$-functions. 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