Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.923 + 0.384i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.338 + 0.940i)3-s + (−0.809 + 0.587i)7-s + (−0.770 − 0.637i)9-s + (0.661 − 0.750i)11-s + (−0.995 + 0.0941i)13-s + (0.125 − 0.992i)17-s + (−0.338 − 0.940i)19-s + (−0.278 − 0.960i)21-s + (−0.929 + 0.368i)23-s + (0.860 − 0.509i)27-s + (0.0314 − 0.999i)29-s + (−0.992 − 0.125i)31-s + (0.481 + 0.876i)33-s + (0.509 − 0.860i)37-s + (0.248 − 0.968i)39-s + ⋯
L(s,χ)  = 1  + (−0.338 + 0.940i)3-s + (−0.809 + 0.587i)7-s + (−0.770 − 0.637i)9-s + (0.661 − 0.750i)11-s + (−0.995 + 0.0941i)13-s + (0.125 − 0.992i)17-s + (−0.338 − 0.940i)19-s + (−0.278 − 0.960i)21-s + (−0.929 + 0.368i)23-s + (0.860 − 0.509i)27-s + (0.0314 − 0.999i)29-s + (−0.992 − 0.125i)31-s + (0.481 + 0.876i)33-s + (0.509 − 0.860i)37-s + (0.248 − 0.968i)39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.923 + 0.384i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.923 + 0.384i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.923 + 0.384i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (3813, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ -0.923 + 0.384i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.04760495701 - 0.2383477839i$
$L(\frac12,\chi)$  $\approx$  $-0.04760495701 - 0.2383477839i$
$L(\chi,1)$  $\approx$  0.6758937511 + 0.04605637839i
$L(1,\chi)$  $\approx$  0.6758937511 + 0.04605637839i

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

−18.68213437867407153607728507155, −17.998525246676905489175187084341, −17.32859353293280789149006896958, −16.62020803647350853144333020698, −16.44442892316217743672919201453, −15.07668179547644201151843832571, −14.55800778488653090095871835022, −13.915103767809921774139106510796, −12.99788297218455364900305936814, −12.5179407785352254805480014465, −12.133208156256166369325108768027, −11.21857974560074036266407576634, −10.23073959064898855261624672482, −9.98095995134914947342322468269, −8.88214904922263536690649675185, −8.09475754865030382933073135111, −7.335561246174247815516263084509, −6.8157532118405374143647973487, −6.16570070971470140076040342583, −5.43365819216994656657907329091, −4.39746175990246001043796914565, −3.69106198620050511314537273624, −2.70366858143706421451258594805, −1.80036303616629842901683134690, −1.18308204967138581072626277377, 0.073567006986375664236243392334, 0.45423579187104943084378067930, 2.07719608549571855262267800833, 2.88316620634649770353167993205, 3.5687827697646647183476291098, 4.35869199947879540213588588677, 5.192268215438410498056368738582, 5.81237436073726955848938863436, 6.51974620101285241610461049075, 7.27688592586522715968249604453, 8.41859091733648215177068316426, 9.07211742872113408061660872550, 9.7034689925381397910229260618, 10.04443095202994762168443785464, 11.27119066244876805670080094226, 11.560385918752713739892526024512, 12.270059978533097295387204981804, 13.140042127579074279688417603054, 13.950248205629555225610636412485, 14.67781072667889698223197010175, 15.27392488026339723516207271834, 16.02211667005784480966319106835, 16.452030081760288242806586871192, 17.087458990859291944684643925658, 17.84660511689252669425334832132

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