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  :  $\chi_{4000} (2237, \cdot )$
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

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

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