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Properties

Label 1-4000-4000.1141-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.0800 + 0.996i$
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  + (0.562 + 0.827i)3-s + (−0.587 − 0.809i)7-s + (−0.368 + 0.929i)9-s + (0.509 + 0.860i)11-s + (0.917 − 0.397i)13-s + (−0.876 + 0.481i)17-s + (0.827 + 0.562i)19-s + (0.338 − 0.940i)21-s + (0.998 − 0.0627i)23-s + (−0.975 + 0.218i)27-s + (−0.612 + 0.790i)29-s + (0.876 − 0.481i)31-s + (−0.425 + 0.904i)33-s + (0.218 − 0.975i)37-s + (0.844 + 0.535i)39-s + ⋯
L(s)  = 1  + (0.562 + 0.827i)3-s + (−0.587 − 0.809i)7-s + (−0.368 + 0.929i)9-s + (0.509 + 0.860i)11-s + (0.917 − 0.397i)13-s + (−0.876 + 0.481i)17-s + (0.827 + 0.562i)19-s + (0.338 − 0.940i)21-s + (0.998 − 0.0627i)23-s + (−0.975 + 0.218i)27-s + (−0.612 + 0.790i)29-s + (0.876 − 0.481i)31-s + (−0.425 + 0.904i)33-s + (0.218 − 0.975i)37-s + (0.844 + 0.535i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.0800 + 0.996i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.0800 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526790901 + 1.409128805i\)
\(L(\frac12)\) \(\approx\) \(1.526790901 + 1.409128805i\)
\(L(1)\) \(\approx\) \(1.216141645 + 0.4248519983i\)
\(L(1)\) \(\approx\) \(1.216141645 + 0.4248519983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.562 + 0.827i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (0.509 + 0.860i)T \)
13 \( 1 + (0.917 - 0.397i)T \)
17 \( 1 + (-0.876 + 0.481i)T \)
19 \( 1 + (0.827 + 0.562i)T \)
23 \( 1 + (0.998 - 0.0627i)T \)
29 \( 1 + (-0.612 + 0.790i)T \)
31 \( 1 + (0.876 - 0.481i)T \)
37 \( 1 + (0.218 - 0.975i)T \)
41 \( 1 + (-0.998 - 0.0627i)T \)
43 \( 1 + (0.891 - 0.453i)T \)
47 \( 1 + (-0.728 + 0.684i)T \)
53 \( 1 + (0.338 - 0.940i)T \)
59 \( 1 + (0.995 - 0.0941i)T \)
61 \( 1 + (0.750 + 0.661i)T \)
67 \( 1 + (-0.790 + 0.612i)T \)
71 \( 1 + (0.684 + 0.728i)T \)
73 \( 1 + (-0.770 + 0.637i)T \)
79 \( 1 + (0.187 - 0.982i)T \)
83 \( 1 + (0.827 + 0.562i)T \)
89 \( 1 + (0.770 - 0.637i)T \)
97 \( 1 + (-0.992 + 0.125i)T \)
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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

−18.4250335377212658567499656804, −17.8776408747105208437064930813, −17.0103771933619362621159869388, −16.21341608071957936142814005945, −15.510206383499634573684972805806, −14.99431243401223897396292356043, −14.007945764643021134767986465768, −13.40064087833721446228652197288, −13.18931093752442103710978131366, −12.03048751747861386115377581649, −11.634406681026324258907618273028, −10.98982081006036643968935788552, −9.70455826703867450370113923996, −9.09359051934488544090373089422, −8.6900912310505034732709581589, −7.96077755126019666828311268462, −6.88388637197379073237290277635, −6.509957261488887199539336212572, −5.8301846254462110413814732525, −4.88009436714076108064908352131, −3.71479219179881231355201095871, −3.09487749216921563170962993593, −2.46750532843184425297129373957, −1.46732992263811311984479918421, −0.62618052280265507455975307970, 0.99719677958259345694364039404, 1.97796683078510021275256432354, 2.98418743959224637529565598505, 3.733128242195628334211063941067, 4.14789470795856460127146956571, 5.03704883020300788035384071401, 5.894900561304957760742125626931, 6.834647922712000108784500181623, 7.42568324117158549579275635556, 8.34276612363823018548039717526, 9.004838580215034117997302983636, 9.68463464597343697178469035501, 10.27605154652394311574852093590, 10.912113836917081028200621557, 11.58247568092039456655311720430, 12.74418383582858122303035148681, 13.231728440933261557646691217070, 13.8884133101749782496486092299, 14.70698206643140117365525595002, 15.153994421369223791646065735894, 16.10319777937330061486548143549, 16.30256174906905009688598410544, 17.331179657775458030217571494653, 17.72209877597348995381132958998, 18.87878910333309644533479518911

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