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  + (−0.509 − 0.860i)3-s + (0.309 + 0.951i)7-s + (−0.481 + 0.876i)9-s + (−0.562 + 0.827i)11-s + (0.278 + 0.960i)13-s + (0.368 − 0.929i)17-s + (−0.509 + 0.860i)19-s + (0.661 − 0.750i)21-s + (−0.425 + 0.904i)23-s + (0.999 − 0.0314i)27-s + (−0.995 − 0.0941i)29-s + (−0.929 − 0.368i)31-s + (0.998 + 0.0627i)33-s + (0.0314 − 0.999i)37-s + (0.684 − 0.728i)39-s + ⋯
L(s,χ)  = 1  + (−0.509 − 0.860i)3-s + (0.309 + 0.951i)7-s + (−0.481 + 0.876i)9-s + (−0.562 + 0.827i)11-s + (0.278 + 0.960i)13-s + (0.368 − 0.929i)17-s + (−0.509 + 0.860i)19-s + (0.661 − 0.750i)21-s + (−0.425 + 0.904i)23-s + (0.999 − 0.0314i)27-s + (−0.995 − 0.0941i)29-s + (−0.929 − 0.368i)31-s + (0.998 + 0.0627i)33-s + (0.0314 − 0.999i)37-s + (0.684 − 0.728i)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  :  $\chi_{4000} (453, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ -0.969 - 0.244i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.09083941121 + 0.7329524862i$
$L(\frac12,\chi)$  $\approx$  $-0.09083941121 + 0.7329524862i$
$L(\chi,1)$  $\approx$  0.7714750829 + 0.1250746929i
$L(1,\chi)$  $\approx$  0.7714750829 + 0.1250746929i

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

−17.84447262306883306122799469289, −17.06601557399574493279764029437, −16.68534973359002579632744690086, −16.02548496670917698658405212767, −15.114944222107741631017982157135, −14.85116174023210987677796126713, −13.798743186448247492608442098495, −13.223671386313397596411526054613, −12.45636474468509466504675335685, −11.55203232725655152744097418732, −10.73781756911175803431438687456, −10.56960611307415024431872191148, −9.93702162854700365820956688708, −8.71657962114076830857125153632, −8.41556602485145944280624603576, −7.4146473443427167967843249995, −6.587439371575908239896449097415, −5.7237210481570866592649760165, −5.23602323548243604479600403563, −4.33576898920474486249732511907, −3.65206059601728781605814934308, −3.06273632283049928496946450071, −1.80527437566016513177368744792, −0.59250208157942812497104056739, −0.17951930794714042194711956071, 1.20088938094805734897596345872, 2.01122469380904024239336572201, 2.39720536767830621953827848705, 3.638458421679435681775283720505, 4.639846574437014134229715621870, 5.435679640859976581701276444725, 5.8759077601410642357011692355, 6.765832958120761349358228473692, 7.563856453968619730715305515147, 7.95328400906195627407123034635, 9.01716947881639460257488308986, 9.54268446576032108303068510556, 10.55595878283257914530391818130, 11.35785602809145979136406524114, 11.85179207694897345865430944411, 12.44138646915241319246437673979, 13.087272606701017396936829927118, 13.84703735268035466143829196040, 14.56802873017562472248245687812, 15.22592281283361580335298260863, 16.15849566961024072218561103717, 16.60747530462117431255990591083, 17.48504041287697403990997212682, 18.19883308972914693687080953994, 18.505876423040851628884480825523

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