Does p-adic $L$- function determines the $L$ functions
Let $E_1$ and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover p-adic $L$-function of $E_1$ and $E_2$ are same . Does it mean that the complex $L$-function of $E_1$ and $E_2$ are also same ?
Is there some sufficient criteria on p-adic $L$-functions such that such that the $L$ function is determined?