in the 1970's stark made precise conjectures about the leading term of the taylor series expansion at $s=0$ of artin $l$-functions, refining dirichlet's class number formula. around the same time barsky, cassou-nogu\`{e}s, and deligne and ribet for totally real fields, along with katz for cm fields defined $p$-adic $l$-functions of ray class characters. since then stark-type conjectures for these $p$-adic $l$-functions have been formulated, and progress has been made in some cases. the goal of this talk is to discuss a new definition of a $p$-adic $l$-function and stark conjecture for a mixed signature character of a real quadratic field. after stating the definition and conjecture, theoretical and numerical evidence will be discussed.