Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.997 + 0.0674i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.975 + 0.218i)3-s + (0.309 − 0.951i)7-s + (0.904 + 0.425i)9-s + (0.0314 − 0.999i)11-s + (−0.338 − 0.940i)13-s + (0.998 − 0.0627i)17-s + (0.975 − 0.218i)19-s + (0.509 − 0.860i)21-s + (−0.187 + 0.982i)23-s + (0.790 + 0.612i)27-s + (0.397 + 0.917i)29-s + (0.0627 + 0.998i)31-s + (0.248 − 0.968i)33-s + (0.612 + 0.790i)37-s + (−0.125 − 0.992i)39-s + ⋯
L(s,χ)  = 1  + (0.975 + 0.218i)3-s + (0.309 − 0.951i)7-s + (0.904 + 0.425i)9-s + (0.0314 − 0.999i)11-s + (−0.338 − 0.940i)13-s + (0.998 − 0.0627i)17-s + (0.975 − 0.218i)19-s + (0.509 − 0.860i)21-s + (−0.187 + 0.982i)23-s + (0.790 + 0.612i)27-s + (0.397 + 0.917i)29-s + (0.0627 + 0.998i)31-s + (0.248 − 0.968i)33-s + (0.612 + 0.790i)37-s + (−0.125 − 0.992i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.997 + 0.0674i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.997 + 0.0674i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.997 + 0.0674i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (317, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ 0.997 + 0.0674i)$
$L(\chi,\frac{1}{2})$  $\approx$  $4.701083664 + 0.1588259543i$
$L(\frac12,\chi)$  $\approx$  $4.701083664 + 0.1588259543i$
$L(\chi,1)$  $\approx$  1.786259648 - 0.06838180918i
$L(1,\chi)$  $\approx$  1.786259648 - 0.06838180918i

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

−18.51167947306016634908371258420, −17.83208727846045315375776682238, −16.952142271199498472895627447047, −16.12563431422826440449045027285, −15.43270863075389813426294233560, −14.8000496977250016354749681748, −14.31763276537495505717174538762, −13.71100616705804721385663091223, −12.71338027218420374926102369883, −12.167032774897958833569219457804, −11.767099139122950833787012725703, −10.582204764066356454206784331217, −9.5699794044686582782253336534, −9.48195461956274475125556488830, −8.55848550962190356090486150866, −7.75816878120642161215005313297, −7.36087106979458009866084030727, −6.37890677013843238758900094851, −5.6096423930905986122907774619, −4.58150479014492852529383797194, −4.07413630044354087234518322049, −2.9880506512908424304578781446, −2.27205533233682550918531092855, −1.79146303422262027534967835400, −0.67912366876628313504544823642, 0.95785057416592347098045627327, 1.214763128062317247018552715986, 2.62110404815737804986673574999, 3.31573705541490975631628877844, 3.6980564659852988567648488728, 4.85126058791013846226610621831, 5.35077222943813193680189462515, 6.462783263864512556444983796756, 7.43637756374292074236498441441, 7.83521481887832393795308236226, 8.40391774071512216248387704096, 9.4556368406083019689533382619, 9.859855864241184483377154068404, 10.723266640423238862684501528890, 11.20684783967939669133973345115, 12.32463780544469136552611302354, 12.96485395731256303707715327631, 13.90513758391950424258621071512, 14.021681149825330870365555155645, 14.76164816820228911254562358530, 15.687285158259024751826694918009, 16.14695650732158235989042926888, 16.87485172083850434360326390644, 17.718162900412311305013956023689, 18.3357899381143918017833175931

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