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Properties

Label 1-4000-4000.1347-r0-0-0
Degree $1$
Conductor $4000$
Sign $-0.444 - 0.895i$
Analytic cond. $18.5759$
Root an. cond. $18.5759$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1390087570 - 0.2240671712i\)
\(L(\frac12)\) \(\approx\) \(0.1390087570 - 0.2240671712i\)
\(L(1)\) \(\approx\) \(0.8400977619 + 0.1804507678i\)
\(L(1)\) \(\approx\) \(0.8400977619 + 0.1804507678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.218 + 0.975i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.999 + 0.0314i)T \)
13 \( 1 + (0.940 + 0.338i)T \)
17 \( 1 + (-0.998 - 0.0627i)T \)
19 \( 1 + (0.218 - 0.975i)T \)
23 \( 1 + (0.187 + 0.982i)T \)
29 \( 1 + (0.917 + 0.397i)T \)
31 \( 1 + (-0.0627 + 0.998i)T \)
37 \( 1 + (0.790 + 0.612i)T \)
41 \( 1 + (-0.982 - 0.187i)T \)
43 \( 1 + (-0.987 - 0.156i)T \)
47 \( 1 + (0.770 - 0.637i)T \)
53 \( 1 + (-0.509 - 0.860i)T \)
59 \( 1 + (-0.960 + 0.278i)T \)
61 \( 1 + (0.562 - 0.827i)T \)
67 \( 1 + (-0.917 + 0.397i)T \)
71 \( 1 + (0.770 - 0.637i)T \)
73 \( 1 + (0.876 - 0.481i)T \)
79 \( 1 + (-0.535 + 0.844i)T \)
83 \( 1 + (0.975 + 0.218i)T \)
89 \( 1 + (0.481 + 0.876i)T \)
97 \( 1 + (-0.368 - 0.929i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.643457796955976218027692906276, −18.229909549311864224435092556921, −17.61362027081717540942355516782, −16.65174847864949995132943235507, −15.91351177468647439556260205306, −15.28368427145850511691045622149, −14.65958401411361566815145251326, −13.67529736691991552281958175259, −13.24623355989020452295203892365, −12.59121868631804860939916444447, −12.03581940635845663460859391996, −11.210542138336663232234151814488, −10.53973948349322221378383626503, −9.57485146809432994215225341093, −8.7512934547456779311135558724, −8.236684955428822740900077826146, −7.69336489181540989905969450631, −6.60116828351029729489611464346, −6.11049143593496681786848565979, −5.5300185988152450757753006072, −4.50690541202471823283901630058, −3.404336453171597588834821330187, −2.6482118218768593578613801710, −2.141831157617218767787965664551, −1.096842238428824301138533613756, 0.07425215993748188908323358431, 1.35806972560492435246295033469, 2.53040634285120422404744334850, 3.28797725221134574178261813373, 3.87772490935922453915797492565, 4.83147008795325330685236574276, 5.16065786705291630818719131784, 6.38784948130385211888513470484, 6.93990857875850728917621937654, 7.940826861285185480631114147498, 8.58450439647961181531611042838, 9.30704969769854954827097696568, 10.00027579267874349763755931432, 10.76232630719436230184531404733, 11.03293036170043681358487656243, 11.91498563089659068704500042669, 13.134184746034102988878207958453, 13.56881440525003741057600705727, 13.98364752501261279206706197137, 15.131036532316720730005389162200, 15.58960616593784603646878119517, 16.11648021931223500863786531541, 16.755618121041607301675349889459, 17.565649574418244791424022669275, 18.10226219411352907704272487076

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