Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k\ge 2$ and of the same level $N$, both eigenfunctions of the Hecke operators, and both normalized so that $a_1\left(f\right)=a_1\left(E\right)=1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\overline{\mathbb{Q}}$ lying over a rational prime $p\>2$), the algebraic parts of the special values $L(E,\chi ,j)$ and $L(f,\chi ,j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions,$$\frac{\tau \left(\overline{\chi }\right)L(f,\chi ,j)}{{\left(2\pi i\right)}^{j-1}{\Omega }_f^{\text{sgn}\left(E\right)}}\equiv \frac{\tau \left(\overline{\chi }\right)L(E,\chi ,j)}{{\left(2\pi i\right)}^j{\Omega }_E}(mod𝔭)$$where the...

Special values of certain $L$ functions of the type $L(M,s)$ are studied where $M$ is a motive over a totally real field $F$ with coefficients in another field $T$, and$$L(M,s)=\prod _{𝔭}{L}_{𝔭}(M,𝒩{𝔭}^{-s})$$is an Euler product $𝔭$ running through maximal ideals of the maximal order ${𝒪}_F$ of $F$ and$$\begin{array}{cc}\hfill L_{𝔭}(M,X{)}^{-1}& =(1-{\alpha }^{\left(1\right)}(𝔭\left)X\right)·(1-{\alpha }^{\left(2\right)}(𝔭\left)X\right)·...·(1-\alpha (d\left)\right(𝔭\left)X\right)\hfill \\ & =1+A_1\left(𝔭\right)X+...+A_d\left(𝔭\right)X^d\hfill \end{array}$$being a polynomial with coefficients in $T$. Using the Newton and the Hodge polygons of $M$ one formulate a conjectural criterium for the existence of a $p$-adic analytic continuation of the special values. This conjecture is verified in a number of cases related to...