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Properties

Label 1-4000-4000.1061-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.890 + 0.455i$
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  + (−0.278 + 0.960i)3-s + (0.587 − 0.809i)7-s + (−0.844 − 0.535i)9-s + (0.397 + 0.917i)11-s + (−0.218 − 0.975i)13-s + (−0.728 + 0.684i)17-s + (0.960 − 0.278i)19-s + (0.612 + 0.790i)21-s + (0.770 + 0.637i)23-s + (0.750 − 0.661i)27-s + (0.827 + 0.562i)29-s + (0.728 − 0.684i)31-s + (−0.992 + 0.125i)33-s + (−0.661 + 0.750i)37-s + (0.998 + 0.0627i)39-s + ⋯
L(s)  = 1  + (−0.278 + 0.960i)3-s + (0.587 − 0.809i)7-s + (−0.844 − 0.535i)9-s + (0.397 + 0.917i)11-s + (−0.218 − 0.975i)13-s + (−0.728 + 0.684i)17-s + (0.960 − 0.278i)19-s + (0.612 + 0.790i)21-s + (0.770 + 0.637i)23-s + (0.750 − 0.661i)27-s + (0.827 + 0.562i)29-s + (0.728 − 0.684i)31-s + (−0.992 + 0.125i)33-s + (−0.661 + 0.750i)37-s + (0.998 + 0.0627i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.890 + 0.455i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1061, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.890 + 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.645009874 + 0.3962986425i\)
\(L(\frac12)\) \(\approx\) \(1.645009874 + 0.3962986425i\)
\(L(1)\) \(\approx\) \(1.053264356 + 0.2191381550i\)
\(L(1)\) \(\approx\) \(1.053264356 + 0.2191381550i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.278 + 0.960i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (0.397 + 0.917i)T \)
13 \( 1 + (-0.218 - 0.975i)T \)
17 \( 1 + (-0.728 + 0.684i)T \)
19 \( 1 + (0.960 - 0.278i)T \)
23 \( 1 + (0.770 + 0.637i)T \)
29 \( 1 + (0.827 + 0.562i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (-0.661 + 0.750i)T \)
41 \( 1 + (-0.770 + 0.637i)T \)
43 \( 1 + (0.453 - 0.891i)T \)
47 \( 1 + (0.425 - 0.904i)T \)
53 \( 1 + (0.612 + 0.790i)T \)
59 \( 1 + (0.509 + 0.860i)T \)
61 \( 1 + (-0.0941 - 0.995i)T \)
67 \( 1 + (-0.562 - 0.827i)T \)
71 \( 1 + (-0.904 - 0.425i)T \)
73 \( 1 + (-0.248 - 0.968i)T \)
79 \( 1 + (-0.876 + 0.481i)T \)
83 \( 1 + (0.960 - 0.278i)T \)
89 \( 1 + (0.248 + 0.968i)T \)
97 \( 1 + (-0.187 - 0.982i)T \)
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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

−18.51766673984565609007297240674, −17.67292862334280156486193555484, −17.39598965182447776422770381275, −16.25079420651162274920315789616, −16.01960465151013650499077912304, −14.84746190666021699483576532294, −14.0940400085208270755055668584, −13.84677800192407016303519310197, −12.913965677336889755848541398581, −12.08498501100847406716606601436, −11.62888495439230390495598613240, −11.22918630094577239415165579058, −10.2415755293196559696396836353, −9.00591100020470400010585784676, −8.799975860928578419063053067938, −7.96784333034657898137489418484, −7.083988737729797639089861326656, −6.55663183016286008624089480562, −5.759913425395146692897613796712, −5.09687942162295076488691547047, −4.31448754944116880403770872395, −3.01645601104053709785026703240, −2.470931916927407849493353065149, −1.56024624045234600559018537733, −0.77856296645731878892719293222, 0.703142404943860185339301988502, 1.6552267505942808029801786341, 2.84995495335143756016444413138, 3.576621885674650047147858697156, 4.414733723334118630548172458208, 4.901997727053370979546739169103, 5.59905268429005653656032740773, 6.63410925605857164661420799478, 7.28886138745626372995390188279, 8.13394921014992717826546321700, 8.89817521978698183467566709701, 9.66465428803178010795216758075, 10.41315689310345920590408918477, 10.67744866826860264170832615462, 11.70207234911097004124762396069, 12.08118041862772447262676359949, 13.21594104564208671454492681087, 13.77659743400102225619226121897, 14.65416763525810642289853371910, 15.30423859008244310733238144169, 15.53712277242012301415899356718, 16.68257834113483205194860875742, 17.19955825128212243307255391831, 17.62841541895694945593236458630, 18.22848947796123330758036041776

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