There are many reasons that functional equations are important. Some background: To most varieties/schemes occurring in arithmetic geometry, you can associate a zeta function/L-function. There are two main ways of constructing these, either from l-adic cohomology, or from counting solutions in various finite fields. Usually the word L-function is used for functions associated to varieties over number fields, constructed from l-adic cohomology, and the word zeta function is used for functions associated to schemes over Z or some other ring of integers, or over a finite field. However, there is some confusion about the terminology, and there is also some overlap between the two, since there is a close relation between the L-function of a variety and the zeta function of an integral model for the variety.

Over finite fields, the functional equation is part of the famous Weil conjectures, proved by Deligne. One reason that the functional equation is cool is that it reflects Poincare duality in the etale cohomology of the variety, so it is in some sense a deep geometric statement. For background on the Weil conjectures, see for example Freitag and Kiehl: Etale cohomology and the Weil conjectures, and the survey of Mazur: Eigenvalues of Frobenius acting on algebraic varieties over finite fields, in some conference proceedings.

For the definition of L-functions and zeta functions and lots of other background, see Manin and Panchishkin: Introduction to modern number theory, chapter 6. There are at least two deep reasons for being interested in the functional equations for these functions. Firstly, the existence of a functional equation seems to always be directly related to the L-function coming from an automorphic representation, and the idea that "every L-function from algebraic geometry (aka motivic L-function) also comes from an automorphic representation" is in some sense the number-theoretic incarnation of the global Langlands program. See for example Bump et al: An introduction to the Langlands program. The most famous cases where this has been proved is (1) Tate's thesis, which treats Hecke L-functions, and where the corresponding automorphic representation is one-dimensional, i.e. a character on the ideles, and (2) the work by Wiles and others related to Fermat's last theorem, where they show that the L-function associated to an elliptic curve over Q also comes from a modular form, and hence satisfies the expected functional equation. See the book of Diamond and Shurman on modular forms.

The other deep reason for thinking about the functional equation is that some optimistic people dream of an "arithmetic cohomology theory", which would allow us to mimick the proof of the Weil conjectures, but for zeta functions over Z or L-functions over Q. Then the functional equation should be related to Poincare duality for this cohomology. All this is related to the Riemann hypothesis, noncommutative geometry, and the field with one element. See for example Deninger: Motivic L-functions and regularized determinants, the more recent survey available here, some slides of Paugam on the functional equation, and also the later chapters of Manin-Panchishkin. Some of the key names if you want to find more references: Deninger, Connes, Consani, Marcolli; most of them have lots of stuff on their webpages and on the arXiv.

There are many reasons that functional equations are important. Some background: To most varieties/schemes occurring in arithmetic geometry, you can associate a zeta function/L-function. There are two main ways of constructing these, either from l-adic cohomology, or from counting solutions in various finite fields. Usually the word L-function is used for functions associated to varieties over number fields, constructed from l-adic cohomology, and the word zeta function is used for functions associated to schemes over Z or some other ring of integers, or over a finite field. However, there is some confusion about the terminology, and there is also some overlap between the two, since there is a close relation between the L-function of a variety and the zeta function of an integral model for the variety.

Over finite fields, the functional equation is part of the famous Weil conjectures, proved by Deligne. One reason that the functional equation is cool is that it reflects Poincare duality in the etale cohomology of the variety, so it is in some sense a deep geometric statement. For background on the Weil conjectures, see for example Freitag and Kiehl: Etale cohomology and the Weil conjectures, and the survey of Mazur: Eigenvalues of Frobenius acting on algebraic varieties over finite fields, in some conference proceedings.

For the definition of L-functions and zeta functions and lots of other background, see Manin and Panchishkin: Introduction to modern number theory, chapter 6. There are at least two deep reasons for being interested in the functional equations for these functions. Firstly, the existence of a functional equation seems to always be directly related to the L-function coming from an automorphic representation, and the idea that "every L-function from algebraic geometry (aka motivic L-function) also comes from an automorphic representation" is in some sense the number-theoretic incarnation of the global Langlands program. See for example Bump et al: An introduction to the Langlands program. The most famous cases where this has been proved is (1) Tate's thesis, which treats Hecke L-functions, and where the corresponding automorphic representation is one-dimensional, i.e. a character on the ideles, and (2) the work by Wiles and others related to Fermat's last theorem, where they show that the L-function associated to an elliptic curve over Q also comes from a modular form, and hence satisfies the expected functional equation. See the book of Diamond and Shurman on modular forms.

The other deep reason for thinking about the functional equation is that some optimistic people dream of an "arithmetic cohomology theory", which would allow us to mimick the proof of the Weil conjectures, but for zeta functions over Z or L-functions over Q. Then the functional equation should be related to Poincare duality for this cohomology. All this is related to the Riemann hypothesis, noncommutative geometry, and the field with one element. See for example Deninger's Motivic L-functions and regularized determinants (pdf), his more recent survey Arithmetic Geometry and Analysis on Foliated Spaces, some slides of Paugam^{link broken} on the functional equation, and also the later chapters of Manin-Panchishkin. Some of the key names if you want to find more references: Deninger, Connes, Consani, Marcolli; most of them have lots of stuff on their webpages and on the arXiv.

There are many reasons that functional equations are important. Some background: To most varieties/schemes occurring in arithmetic geometry, you can associate a zeta function/L-function. There are two main ways of constructing these, either from l-adic cohomology, or from counting solutions in various finite fields. Usually the word L-function is used for functions associated to varieties over number fields, constructed from l-adic cohomology, and the word zeta function is used for functions associated to schemes over Z or some other ring of integers, or over a finite field. However, there is some confusion about the terminology, and there is also some overlap between the two, since there is a close relation between the L-function of a variety and the zeta function of an integral model for the variety.

Over finite fields, the functional equation is part of the famous Weil conjectures, proved by Deligne. One reason that the functional equation is cool is that it reflects Poincare duality in the etale cohomology of the variety, so it in some sense a deep geometric statement. For background on the Weil conjectures, see for example Freitag and Kiehl: Etale cohomology and the Weil conjectures, and the survey of Mazur: Eigenvalues of Frobenius acting on algebraic varieties over finite fields, in some conference proceedings.

For the definition of L-functions and zeta functions and lots of other background, see Manin and Panchishkin: Introduction to modern number theory, chapter 6. There are at least two deep reasons for being interested in the functional equations for these functions. Firstly, the existence of a functional equation seems to always be directly related to the L-function coming from an automorphic representation, and the idea that "every L-function from algebraic geometry (aka motivic L-function) also comes from an automorphic representation" is in some sense the number-theoretic incarnation of the global Langlands program. See for example Bump et al: An introduction to the Langlands program. The most famous cases where this has been proved is (1) Tate's thesis, which treats Hecke L-functions, and where the corresponding automorphic representation is one-dimensional, i.e. a character on the ideles, and (2) the work by Wiles and others related to Fermat's last theorem, where they show that the L-function associated to an elliptic curve over Q also comes from a modular form, and hence satisfies the expected functional equation. See the book of Diamond and Shurman on modular forms.

The other deep reason for thinking about the functional equation is that some optimistic people dream of a "arithmetic cohomology theory", which would allow us to mimick the proof of the Weil conjectures, but for zeta functions over Z or L-functions over Q. Then the functional equation should be related to Poincare duality for this cohomology. All this is related to the Riemann hypothesis, noncommutative geometry, and the field with one element. See for example Deninger: Motivic L-functions and regularized determinants, and the more recent survey available here.

There are many reasons that functional equations are important. Some background: To most varieties/schemes occurring in arithmetic geometry, you can associate a zeta function/L-function. There are two main ways of constructing these, either from l-adic cohomology, or from counting solutions in various finite fields. Usually the word L-function is used for functions associated to varieties over number fields, constructed from l-adic cohomology, and the word zeta function is used for functions associated to schemes over Z or some other ring of integers, or over a finite field. However, there is some confusion about the terminology, and there is also some overlap between the two, since there is a close relation between the L-function of a variety and the zeta function of an integral model for the variety.

Over finite fields, the functional equation is part of the famous Weil conjectures, proved by Deligne. One reason that the functional equation is cool is that it reflects Poincare duality in the etale cohomology of the variety, so it is in some sense a deep geometric statement. For background on the Weil conjectures, see for example Freitag and Kiehl: Etale cohomology and the Weil conjectures, and the survey of Mazur: Eigenvalues of Frobenius acting on algebraic varieties over finite fields, in some conference proceedings.

For the definition of L-functions and zeta functions and lots of other background, see Manin and Panchishkin: Introduction to modern number theory, chapter 6. There are at least two deep reasons for being interested in the functional equations for these functions. Firstly, the existence of a functional equation seems to always be directly related to the L-function coming from an automorphic representation, and the idea that "every L-function from algebraic geometry (aka motivic L-function) also comes from an automorphic representation" is in some sense the number-theoretic incarnation of the global Langlands program. See for example Bump et al: An introduction to the Langlands program. The most famous cases where this has been proved is (1) Tate's thesis, which treats Hecke L-functions, and where the corresponding automorphic representation is one-dimensional, i.e. a character on the ideles, and (2) the work by Wiles and others related to Fermat's last theorem, where they show that the L-function associated to an elliptic curve over Q also comes from a modular form, and hence satisfies the expected functional equation. See the book of Diamond and Shurman on modular forms.

The other deep reason for thinking about the functional equation is that some optimistic people dream of an "arithmetic cohomology theory", which would allow us to mimick the proof of the Weil conjectures, but for zeta functions over Z or L-functions over Q. Then the functional equation should be related to Poincare duality for this cohomology. All this is related to the Riemann hypothesis, noncommutative geometry, and the field with one element. See for example Deninger: Motivic L-functions and regularized determinants, the more recent survey available here, some slides of Paugam on the functional equation, and also the later chapters of Manin-Panchishkin. Some of the key names if you want to find more references: Deninger, Connes, Consani, Marcolli; most of them have lots of stuff on their webpages and on the arXiv.