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Properties

Label 1-4000-4000.1363-r0-0-0
Degree $1$
Conductor $4000$
Sign $-0.452 + 0.891i$
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  + (−0.0314 + 0.999i)3-s + (−0.809 + 0.587i)7-s + (−0.998 − 0.0627i)9-s + (−0.218 − 0.975i)11-s + (0.661 + 0.750i)13-s + (−0.904 − 0.425i)17-s + (0.0314 + 0.999i)19-s + (−0.562 − 0.827i)21-s + (0.968 + 0.248i)23-s + (0.0941 − 0.995i)27-s + (0.278 + 0.960i)29-s + (0.425 − 0.904i)31-s + (0.982 − 0.187i)33-s + (0.995 − 0.0941i)37-s + (−0.770 + 0.637i)39-s + ⋯
L(s)  = 1  + (−0.0314 + 0.999i)3-s + (−0.809 + 0.587i)7-s + (−0.998 − 0.0627i)9-s + (−0.218 − 0.975i)11-s + (0.661 + 0.750i)13-s + (−0.904 − 0.425i)17-s + (0.0314 + 0.999i)19-s + (−0.562 − 0.827i)21-s + (0.968 + 0.248i)23-s + (0.0941 − 0.995i)27-s + (0.278 + 0.960i)29-s + (0.425 − 0.904i)31-s + (0.982 − 0.187i)33-s + (0.995 − 0.0941i)37-s + (−0.770 + 0.637i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.452 + 0.891i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6757398079 + 1.100765620i\)
\(L(\frac12)\) \(\approx\) \(0.6757398079 + 1.100765620i\)
\(L(1)\) \(\approx\) \(0.8504254631 + 0.4021056507i\)
\(L(1)\) \(\approx\) \(0.8504254631 + 0.4021056507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.0314 + 0.999i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.218 - 0.975i)T \)
13 \( 1 + (0.661 + 0.750i)T \)
17 \( 1 + (-0.904 - 0.425i)T \)
19 \( 1 + (0.0314 + 0.999i)T \)
23 \( 1 + (0.968 + 0.248i)T \)
29 \( 1 + (0.278 + 0.960i)T \)
31 \( 1 + (0.425 - 0.904i)T \)
37 \( 1 + (0.995 - 0.0941i)T \)
41 \( 1 + (0.248 + 0.968i)T \)
43 \( 1 + (0.891 - 0.453i)T \)
47 \( 1 + (0.125 + 0.992i)T \)
53 \( 1 + (0.827 - 0.562i)T \)
59 \( 1 + (-0.917 + 0.397i)T \)
61 \( 1 + (-0.509 - 0.860i)T \)
67 \( 1 + (0.278 - 0.960i)T \)
71 \( 1 + (-0.125 - 0.992i)T \)
73 \( 1 + (0.929 - 0.368i)T \)
79 \( 1 + (-0.728 + 0.684i)T \)
83 \( 1 + (0.999 - 0.0314i)T \)
89 \( 1 + (-0.368 - 0.929i)T \)
97 \( 1 + (0.481 - 0.876i)T \)
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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

−18.166575609445024740931039363914, −17.56070783900944962058160680491, −17.218429057462272609992104157900, −16.25866720835667686681025311043, −15.43724176675334824119464617895, −14.99670486359144286911004717939, −13.827778484584356995605729105230, −13.43267811142407603006277060256, −12.79263958508801792604403343913, −12.401219868059148630016843454106, −11.33832693362279394778303755946, −10.7622393866504700821981217225, −10.04010140147769602779359470788, −9.08474060119582515720314134663, −8.50831340540175374099391333093, −7.55857739280186252568813391496, −7.03957500955954785486109475361, −6.44783535890878991804444552812, −5.73034305002055549997039865943, −4.73738731473588645357695916223, −3.93881461176546231450094122287, −2.85097320196655908239891390086, −2.42613567025228533830303132884, −1.237591156074651684008062366620, −0.492225871452992169006819212894, 0.84675876288469345400196384619, 2.23059784715563040489734306951, 3.03851335621269415784567011909, 3.59466687583717438918834871103, 4.439336622912971283055388517050, 5.21868850271799320194659143407, 6.183337915375665185933872481202, 6.302851826955501801011366700474, 7.638261861022801276724753236356, 8.536894319110501640091767309219, 9.13393207251278974982201256896, 9.50469541143356356308174515022, 10.49243452363135498923129445619, 11.11565111590413773794597682292, 11.6086979543903658173440117278, 12.534131298436829044960326439500, 13.34898402435951045967673672161, 13.9595662862926222883112560925, 14.7142236652508798669828919705, 15.52008763353928444524555563108, 15.96048064029659818951367898755, 16.56083412980641568045248482193, 17.019219050391339003413204004924, 18.226147574142520118093306067313, 18.61948987552178167277086012108

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