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Properties

Label 1-4000-4000.1003-r0-0-0
Degree $1$
Conductor $4000$
Sign $-0.641 - 0.767i$
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  + (0.661 − 0.750i)3-s + (−0.309 − 0.951i)7-s + (−0.125 − 0.992i)9-s + (0.940 − 0.338i)11-s + (0.790 + 0.612i)13-s + (−0.770 − 0.637i)17-s + (0.661 + 0.750i)19-s + (−0.917 − 0.397i)21-s + (−0.876 + 0.481i)23-s + (−0.827 − 0.562i)27-s + (0.218 + 0.975i)29-s + (0.637 − 0.770i)31-s + (0.368 − 0.929i)33-s + (−0.562 − 0.827i)37-s + (0.982 − 0.187i)39-s + ⋯
L(s)  = 1  + (0.661 − 0.750i)3-s + (−0.309 − 0.951i)7-s + (−0.125 − 0.992i)9-s + (0.940 − 0.338i)11-s + (0.790 + 0.612i)13-s + (−0.770 − 0.637i)17-s + (0.661 + 0.750i)19-s + (−0.917 − 0.397i)21-s + (−0.876 + 0.481i)23-s + (−0.827 − 0.562i)27-s + (0.218 + 0.975i)29-s + (0.637 − 0.770i)31-s + (0.368 − 0.929i)33-s + (−0.562 − 0.827i)37-s + (0.982 − 0.187i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.641 - 0.767i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.641 - 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8903799728 - 1.903784255i\)
\(L(\frac12)\) \(\approx\) \(0.8903799728 - 1.903784255i\)
\(L(1)\) \(\approx\) \(1.181601753 - 0.6230234432i\)
\(L(1)\) \(\approx\) \(1.181601753 - 0.6230234432i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.661 - 0.750i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.940 - 0.338i)T \)
13 \( 1 + (0.790 + 0.612i)T \)
17 \( 1 + (-0.770 - 0.637i)T \)
19 \( 1 + (0.661 + 0.750i)T \)
23 \( 1 + (-0.876 + 0.481i)T \)
29 \( 1 + (0.218 + 0.975i)T \)
31 \( 1 + (0.637 - 0.770i)T \)
37 \( 1 + (-0.562 - 0.827i)T \)
41 \( 1 + (0.481 - 0.876i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + (0.248 - 0.968i)T \)
53 \( 1 + (0.397 - 0.917i)T \)
59 \( 1 + (-0.999 - 0.0314i)T \)
61 \( 1 + (0.278 - 0.960i)T \)
67 \( 1 + (-0.218 + 0.975i)T \)
71 \( 1 + (0.248 - 0.968i)T \)
73 \( 1 + (0.728 + 0.684i)T \)
79 \( 1 + (-0.0627 - 0.998i)T \)
83 \( 1 + (0.750 - 0.661i)T \)
89 \( 1 + (0.684 - 0.728i)T \)
97 \( 1 + (-0.844 - 0.535i)T \)
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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

−18.843797561817543257096011952249, −18.01989232464919086572559770338, −17.40573658700885829737984024617, −16.5432179164818782537509513779, −15.71191638530607061213293341864, −15.43972441620841810229469368767, −14.85415795503954876025494215085, −13.88486053291919666713150170541, −13.51591414845967869436399937972, −12.51866919759759106317742327568, −11.89151420306904062373182723916, −11.07279001086466822634864471445, −10.3556348401721223558666525764, −9.607250514338003349836562921971, −9.014601730265105778188122204229, −8.46273912995321892412116221468, −7.81442533483832145918596075572, −6.61757413108530406368050279506, −6.0924416494728154381609928902, −5.16962422767890776669525863795, −4.38264582386754119877731979198, −3.70837330678869854077407069892, −2.871744932963648931502261862736, −2.252502732481155218992631705913, −1.22037939132964778006334667107, 0.54977858966216312322927627282, 1.41608377124398827476650965037, 2.0781414701821439295763034402, 3.30845836513511822968514159402, 3.71071162303429983010682280101, 4.44474024985313379753140077447, 5.7578002695845026776612050556, 6.46873665335748623881929113860, 6.99519408483421945110865589809, 7.68532757704998304069097830655, 8.47385535898667679740734043771, 9.18610340447654689974711874789, 9.71515087075750465523107124693, 10.69035831906378079933486137252, 11.54143872758518364811653557713, 12.00908339067461292643222327506, 12.951564705426643182928447878713, 13.60635231659434022327130397078, 14.0737642221294356541005578494, 14.45547452856339704068072263864, 15.61882463361491827028275474042, 16.197056973847527900770987659661, 16.88591738621205899862226515417, 17.73246765570262409790157045300, 18.21942332544833637179971836751

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