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.