Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.452 - 0.891i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.612 − 0.790i)3-s + (−0.809 − 0.587i)7-s + (−0.248 + 0.968i)9-s + (−0.995 + 0.0941i)11-s + (−0.509 − 0.860i)13-s + (−0.982 − 0.187i)17-s + (−0.612 + 0.790i)19-s + (0.0314 + 0.999i)21-s + (0.535 + 0.844i)23-s + (0.917 − 0.397i)27-s + (−0.338 − 0.940i)29-s + (−0.187 + 0.982i)31-s + (0.684 + 0.728i)33-s + (0.397 − 0.917i)37-s + (−0.368 + 0.929i)39-s + ⋯
L(s,χ)  = 1  + (−0.612 − 0.790i)3-s + (−0.809 − 0.587i)7-s + (−0.248 + 0.968i)9-s + (−0.995 + 0.0941i)11-s + (−0.509 − 0.860i)13-s + (−0.982 − 0.187i)17-s + (−0.612 + 0.790i)19-s + (0.0314 + 0.999i)21-s + (0.535 + 0.844i)23-s + (0.917 − 0.397i)27-s + (−0.338 − 0.940i)29-s + (−0.187 + 0.982i)31-s + (0.684 + 0.728i)33-s + (0.397 − 0.917i)37-s + (−0.368 + 0.929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.452 - 0.891i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.452 - 0.891i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.452 - 0.891i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (1173, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ -0.452 - 0.891i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.09104746135 - 0.1483143572i$
$L(\frac12,\chi)$  $\approx$  $0.09104746135 - 0.1483143572i$
$L(\chi,1)$  $\approx$  0.5376693751 - 0.1302573853i
$L(1,\chi)$  $\approx$  0.5376693751 - 0.1302573853i

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

−18.52400361211660420872885008188, −17.871865939330791320784485330205, −16.80874942679109317569480757764, −16.69491154015167226492616001765, −15.759709598222434561621398881283, −15.24141968540534837511690351202, −14.80261074753373928803260461501, −13.65539673199786604068990978549, −12.8971392314327273379209080066, −12.45889222829467412322742682452, −11.41412599181742837128548978085, −11.08625495134669546218846357734, −10.17607897232075647605232320988, −9.62630167798609364845216584945, −8.90823617198474382523605453476, −8.34616517559337799267710697114, −6.98582503428166379381651705537, −6.56409446673797126300115164803, −5.81904521344169117116196568463, −4.84423724191599432526915658050, −4.58682396101009360053763947950, −3.43631291424996007405222877947, −2.74435078319067341399261315772, −1.91664848074899465188408707426, −0.318138093572954847106190789021, 0.08291724501247618798716716128, 1.012480410648022917729265769410, 2.087929859387364523882518265265, 2.774127960312912485750611011675, 3.703953030432577078670128087543, 4.72630026983231627610134347108, 5.45140474200071563498633609838, 6.10445136207387987026401090081, 6.92006535733918831618546250125, 7.48108742482729612184062309410, 8.07377317681540978129007318802, 9.03161222724106034855882943354, 10.04609099231169080563330605845, 10.51818955976196276272889545094, 11.13660504546756241210865186994, 12.06475470485777227054446623716, 12.76096148931484640452164035907, 13.16415123520484718434142488363, 13.69540830424863762818290663624, 14.65796522982710270281763626926, 15.62477822972295711511543167025, 15.998732016129299988345809591258, 16.99366991412316045245344933626, 17.34934613449065298362954731684, 18.0480136569855226179763268954

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