There are many theories that say that a language is simply a product of the culture that speaks it, the same way it's art and food are, so there's no reason to say that yours isn't. However, some linguistic theories claim that language is not influenced much by culture at all. A few natlangs do seem to be influenced by culture; Japanese is the most famous example. The Japanese have throughout history been very hierarchy-conscious. Their verbs inflect to show deference between speaker and listener. Feel free to use your own judgement in how your language and culture interact.
Esperanto: a Culture-Killer, a Culture-Creator, or Culturally Neutral?
As the undisputedly widest-spoken conlang in the world, Esperanto becomes a key point of argument for discussing conlangs in general. Critics of Esperanto have discussed whether or not Esperanto helps or hinders other languages, and it is often feared that if Esperanto was more widely-spoken, people would cease to speak their own languages, resulting in a massive number of languages going extinct. Esperantists have argued many times that Esperanto, as Zamenhof designed it, has not replaced any languages but rather supplemented them as a second language, and they point out that they are one of the few linguistic communities in the world to have virtually all members being multilingual. 
Another question, however, was whether or not Esperanto has a culture or not, and the Esperantists themselves are split on this. Raumism is a political ideology in the Esperanto community which holds that Esperanto, rather than being a neutral language purely for second-language communication between peoples, has evolved its own culture and community in its own rights. Traditional Esperantists continue to argue that Esperanto has only created an interculture, a cultural bridge between different pre-existing cultures, and that Esperanto music and literature are only parts of this rather than being distinct cultural entities in their own rights. At the same time, all Esperantists argue against general critics of Esperanto, who argue that Esperanto lacks a culture and is therefore "not useful"; the traditional Esperantist response is that Esperanto culture exists only as an element of a far larger international culture, while Raumists argue that Esperanto has its own unique culture.
Language (Re-)Creation and Cultural Revival: A Look at Hebrew
The closest analogy that can be found to a constructed language is a language which has been or is being revived. The best example of this by far is Hebrew, which is the only language to have been revived from being unspoken and used only for Jewish liturgy to being the everyday first language of millions of speakers. As more Jews arrived in Mandatory Palestine, dropping the languages they spoke in the diaspora like Yiddish and Judeo-Spanish became a more popular trend to "negate the diaspora", an attempt to erase the time the Jews had been living outside of the land - the phrase "Hebrew [man], speak Hebrew" was common as the politico-cultural connotations of speaking the historical Jewish language conflicted with the "foreign" languages spoken by the Jews when they arrived.
However, much like a language created a priori, Hebrew managed to pick up elements of the other languages of its revivalists. Ghil'ad Zuckermann, a professor of revival linguistics, described the Founder Theory, arguing that a language that is revived will always be reconstructed with inescapable elements of the revivalists' mother languages. According to his theory, Hebrew formed as a hybrid with elements of both Semitic Biblical Hebrew and Indo-European Yiddish, the native language of the early revivalists. This theory equally applies to both reviving and constructing languages: that no matter how careful one is, elements of the language creator's culture will always be present in a conlang in one way or another.
- Manifesto de Prago de la movado por la internacia lingvo Esperanto
- See p. 63 in Zuckermann, Ghil'ad (2006). "A New Vision for 'Israeli Hebrew': Theoretical and Practical Implications of Analysing Israel's Main Language as a Semi-Engineered Semito-European Hybrid Language". Journal of Modern Jewish Studies 5 (1): 57–71. doi:10.1080/14725880500511175. http://www.zuckermann.org/pdf/new-vision.pdf. Retrieved September 19, 2014.