Behavior of quadratic_L_function__exact()

asked 2017-02-09 07:00:24 +0200

user101214 gravatar image

updated 2017-02-09 07:01:37 +0200

I need to evaluate $L$-functions of the form $$\sum_{n=0}^{\infty} \Big( \frac{D}{n} \Big) n^{-s}$$ at integers, where $\Big( \frac{\cdot}{\cdot} \Big)$ is the Kronecker symbol. This is apparently what quadratic_L_function__exact(k,D) does. However I find this unreliable when D is not squarefree.

For example, the value quadratic_L_function__exact(2,4) = pi^2 / 6 is not what I need; the series above is actually pi^2 / 8. However, the value quadratic_L_function__exact(2,12) = 1/18 * sqrt(3) * pi^2 seems correct. I can't tell what's going on.

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It looks like the first operation done is to make D squarefree...

sage: set( [ quadratic_L_function__exact( 2,k^2 ) for k in [1..100] ] )
dan_fulea gravatar imagedan_fulea ( 2017-03-15 19:46:37 +0200 )edit