Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.996 - 0.0800i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.827 − 0.562i)3-s + (−0.809 − 0.587i)7-s + (0.368 + 0.929i)9-s + (0.509 − 0.860i)11-s + (0.397 − 0.917i)13-s + (0.481 − 0.876i)17-s + (−0.827 + 0.562i)19-s + (0.338 + 0.940i)21-s + (0.0627 − 0.998i)23-s + (0.218 − 0.975i)27-s + (0.612 + 0.790i)29-s + (0.876 + 0.481i)31-s + (−0.904 + 0.425i)33-s + (−0.975 + 0.218i)37-s + (−0.844 + 0.535i)39-s + ⋯
L(s,χ)  = 1  + (−0.827 − 0.562i)3-s + (−0.809 − 0.587i)7-s + (0.368 + 0.929i)9-s + (0.509 − 0.860i)11-s + (0.397 − 0.917i)13-s + (0.481 − 0.876i)17-s + (−0.827 + 0.562i)19-s + (0.338 + 0.940i)21-s + (0.0627 − 0.998i)23-s + (0.218 − 0.975i)27-s + (0.612 + 0.790i)29-s + (0.876 + 0.481i)31-s + (−0.904 + 0.425i)33-s + (−0.975 + 0.218i)37-s + (−0.844 + 0.535i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.996 - 0.0800i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.996 - 0.0800i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.996 - 0.0800i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (1277, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ -0.996 - 0.0800i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.05644302039 - 1.408373473i$
$L(\frac12,\chi)$  $\approx$  $0.05644302039 - 1.408373473i$
$L(\chi,1)$  $\approx$  0.7093486229 - 0.3861542922i
$L(1,\chi)$  $\approx$  0.7093486229 - 0.3861542922i

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

−18.724924353636809207766430591602, −17.736007800040710921387635985865, −17.21500806209758802850106475986, −16.73084187445971162653821328782, −15.80775862377666058389144760385, −15.42275301856567268744402299953, −14.82989496411473251931184592675, −13.85222412579284081526803466373, −13.01126252294711727386143506692, −12.29667345463573332002638747218, −11.85053113529974316218228871713, −11.139235297976567352980162951442, −10.248095687784372309465903651451, −9.72400795151360638449316280228, −9.11711228292991701608138611517, −8.38712373988443457955706016754, −7.150175137445868764856888697884, −6.54032236039141177363112881214, −6.02121200927955840466394727244, −5.24877728179712010985175067398, −4.22444627282793825268529054877, −3.94019864437343563575625498648, −2.82656944966042273603209601649, −1.831111859139392459464628994601, −0.90320485177265946679430367817, 0.40033012073226376698500678925, 0.68876123402543280949151719296, 1.71740718156230962370371941029, 2.90876711999711250000775511971, 3.52362722722948233361709794882, 4.51365393830859720937283953987, 5.37200791807375156637066196800, 6.06058935648193375927432493972, 6.705569681905962866642709740140, 7.201104249705228226359545875534, 8.266509171212435165496597544091, 8.72038715090614239713661423613, 9.99749242466937187556252155660, 10.443277198233987671277239425826, 10.98485807200087355975139372935, 12.01838246008497869219943459015, 12.36995628272654332277969115620, 13.16941694975390484701332930487, 13.794829293759280770382849487562, 14.31642786625859180272498404915, 15.56690823118019546249659607496, 16.040457672404308448240465110547, 16.89533875569494077281129092174, 17.01146944227657002639301821820, 18.041294366013680728975047602132

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