They are less familiar with Rick Lazio.
The two rivals tried to close the perception gap Wednesday during an acrimonious debate, the first in the nation's most watched and expensive Senate race.
Mrs. Clinton, whose move to New York just last year remains an election issue, said length of residency should not be the test.
"But if you want someone who will get up every day and be on your side and fight for better schools, health care and jobs, I can pass that test," she said.
Mr. Lazio, whose support is based as much on opposition to her as support for him, told voters in a statewide telecast that "you've got to decide in this campaign how you define character and trust."
"My opponent has talked and talked, but she has done nothing for New York," the four-term House member said. "I've delivered for New York."
How well each candidate defines the other could mean the difference between a new political career for the first lady and a fame-making win for the lesser-known Long Island congressman, pollsters and political analysts said.
"Everybody has an opinion of her, for better or for worse," said Maurice Carroll, director of the Quinnipiac University Polling Institute. "But he's still a bit of a work in progress, and he'll have to show he can be in the big leagues."
During the one-hour showdown at a public television station, Mrs, Clinton repeatedly tried to link Mr. Lazio with former House Speaker Newt Gingrich, a conservative Republican who remains unpopular in this primarily Democratic state.
Citing the 1995 government shutdown and attempted cuts in education and health care, Mrs. Clinton mocked Mr. Lazio's claim to be a mainstream moderate.
"Listening to the congressman's response reminds me of a word I've heard a lot of this past year: chutzpah," Mrs. Clinton said. "Time and time again, when he's had a choice to make ... he stood with the Republican leadership and Newt Gingrich."
Saying his record had been distorted, Mr. Lazio accused Mrs. Clinton of trying to redefine the word chutzpah, telling her that "you of all people shouldn't try to make guilt by association."
"Newt Gingrich isn't running in this race; I'm running in this race," Mr. Lazio said. "Let's talk about my record."
Mrs. Clinton later fielded a question about her husband's impeachment over sworn statements he made about his relationship with Monica Lewinsky. The first lady said she regretted the entire matter, including her initial statements that the accusations against her husband stemmed from a "vast right-wing conspiracy."
"Obviously, I didn't mislead anyone â€“ I didn't know the truth," Mrs. Clinton said. "And there's a great deal of pain associated with that."
Mr. Lazio stood by comments in a fund-raising letter that the first lady "embarrassed our country."
At one point, Mr. Lazio approached Mrs. Clinton's lectern and asked her to sign a pledge to forgo unregulated soft money, which the parties and outside groups often spend on television commercials attacking candidates.
Mrs. Clinton chuckled at Mr. Lazio's offer, noting that conservative groups across the country have pledged millions to help defeat her.
"You know, I admire that â€“ that was a wonderful performance and you did it very well," Mrs. Clinton said.
Countering that liberal groups are spending millions on her behalf, Mr. Lazio said to Mrs. Clinton: "Why don't you show some leadership? Because it goes to truth and character."
The two candidates left the stage without shaking hands.
What the polls say
Mrs. Clinton headed into the showdown with a slight lead, according to two newly released polls.
A Quinnipiac poll put Mrs. Clinton ahead 49 to 44 percent among likely voters, with a margin of error of plus or minus 3.5 percentage points. That survey also showed that, four months after Mr. Lazio entered the race, 19 percent of New Yorkers still don't know enough about him to have a favorable or unfavorable opinion.
Only 2 percent said they don't know enough about Mrs. Clinton.
Another poll this week by the Marist Institute for Public Opinion reflected a closer race among likely voters. Mrs. Clinton led that poll 48 to 46 percent, a statistical dead heat given the error margin of plus or minus 4.5 percentage points.
Marist also reported that 53 percent of Mr. Lazio's backing comes from those who say they oppose her more than they support him.
"She's inching up," said Lee Miringoff, who conducted the poll for Marist. As for Mr. Lazio: "I don't think he's done a whole lot to define himself at this point."
But Utica-based pollster John Zogby said he believes Mrs. Clinton "has reached her peak" and that "this race is Rick Lazio's to lose."
"The issue is, Rick Lazio has to answer two important questions: Who are you, and why should you be our senator?" Mr. Zogby said.
The race to replace retiring Sen. Daniel Patrick Moynihan once featured two well-known personalities, but New York Mayor Rudolph Giuliani withdrew in May amid health and marital problems.
Though national interested waned somewhat after Mr. Lazio took over, the two campaigns estimate that the race has cost more than $60 million already, making it the most expensive Senate contest in history.
Mrs. Clinton's effort is not unprecedented. In 1964, former Attorney General Robert F. Kennedy moved to New York and was elected to the Senate â€“ thanks in part to President Lyndon Johnson's landslide victory.
Though polls reflect strong New York support for Democratic nominee Al Gore, some analysts say the presidential race may not make as much difference this year, given Mrs. Clinton's name recognition.
Mr. Lazio has been accused by some Democrats of basing his campaign on the notion that he is all that's standing between Mrs. Clinton and the Senate. Some of his supporters, however, said he is forging his own identity by calling for tax cuts, school vouchers and teacher testing.
"Do I know a whole lot about Lazio? No," said John Thomas, a marketer from Albany. "But I do know enough about him. ... He has four terms in Congress. He has cast more votes than Hillary Clinton has, and he's a New Yorker."
In addition to challenging her New York credentials, some Lazio backers continue to question her roles in the Whitewater land development and her statements about the firings in the White House travel office. They also suspect that she is using the Senate race as a platform for a future White House run.
Mr. Lazio has repeatedly accused the White House of assisting the first lady, most recently by releasing a photograph of him shaking hands with Palestinian leader Yasser Arafat during a diplomatic trip in 1998.
Mrs. Clinton said the photo exposed the hypocrisy of Mr. Lazio, who had criticized her for embracing Mr. Arafat's wife last fall â€“ a display that stirred the passions of New York state's large Jewish population.
Clinton allies said that while the Republicans talk about scandal and carpetbagging, the first lady has visited all 62 counties in New York and developed a specific agenda.
"We have to look at what she's going to do in the Senate," said Peter Kauffmann, a spokesman for the New York State Democratic Party. "I think people will look at this election and say, 'Who is going to be in my best interest?'"
Staff writer Carl P. Leubsdorf contributed to this report.