Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.969 + 0.244i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.917 − 0.397i)3-s + (0.309 − 0.951i)7-s + (0.684 − 0.728i)9-s + (−0.960 + 0.278i)11-s + (0.999 − 0.0314i)13-s + (−0.844 − 0.535i)17-s + (0.917 + 0.397i)19-s + (−0.0941 − 0.995i)21-s + (−0.992 + 0.125i)23-s + (0.338 − 0.940i)27-s + (0.860 + 0.509i)29-s + (0.535 − 0.844i)31-s + (−0.770 + 0.637i)33-s + (−0.940 + 0.338i)37-s + (0.904 − 0.425i)39-s + ⋯
L(s,χ)  = 1  + (0.917 − 0.397i)3-s + (0.309 − 0.951i)7-s + (0.684 − 0.728i)9-s + (−0.960 + 0.278i)11-s + (0.999 − 0.0314i)13-s + (−0.844 − 0.535i)17-s + (0.917 + 0.397i)19-s + (−0.0941 − 0.995i)21-s + (−0.992 + 0.125i)23-s + (0.338 − 0.940i)27-s + (0.860 + 0.509i)29-s + (0.535 − 0.844i)31-s + (−0.770 + 0.637i)33-s + (−0.940 + 0.338i)37-s + (0.904 − 0.425i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.969 + 0.244i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.969 + 0.244i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.969 + 0.244i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (717, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ -0.969 + 0.244i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1824158086 - 1.471851464i$
$L(\frac12,\chi)$  $\approx$  $-0.1824158086 - 1.471851464i$
$L(\chi,1)$  $\approx$  1.245397877 - 0.4708105037i
$L(1,\chi)$  $\approx$  1.245397877 - 0.4708105037i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.7092843935304327407149997036, −18.018542891095063886499937983008, −17.6306635642675012363331207364, −16.14043833498351676253920139960, −15.86936797797781548707490542375, −15.49969166965968639649012914145, −14.53036556116914635365474917052, −14.01679630447419482250733256010, −13.22686841318933510763547399329, −12.774911098948559782167353533767, −11.69040642757566278263530697445, −11.10680216974733321165872755631, −10.274666211157796777642612396941, −9.663264705118399059048362802100, −8.76079779080386663586429847248, −8.32267609301676842300074984906, −7.872885792833821955222059234618, −6.71059979082843229016078085508, −5.962647775604299911648514115820, −5.06077925505266762370080313781, −4.49529994462285179195317038329, −3.448956568086122572958308451874, −2.86021415394737993935400685942, −2.10320035163035051776054251898, −1.297656610978074028336573401030, 0.173703295228030453694381972900, 1.11532715408034984649024235824, 1.87764071766010552567077048397, 2.73849373164841263103138565122, 3.56737926763627070964177600658, 4.19493070417192477540704225674, 5.052063057316664235155605129823, 6.03874468038305483613113265842, 7.01014162841756089883200559205, 7.37009039402746678720887204879, 8.31981123763316525228849923982, 8.555689044482109132709070998010, 9.79748112307729842265828587059, 10.12816607666338698644509971159, 11.030089487677081449170910627219, 11.7904537921901344439239322816, 12.655605811865410289164468504341, 13.38909471052089826509823187522, 13.84272971025066034037349415087, 14.23646280247296323691545152314, 15.356208599300152191146950245804, 15.77525412301806374317883678823, 16.403610125498706936826952648447, 17.65699129069645933010199469514, 17.889798434970034693048566944814

Graph of the $Z$-function along the critical line