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Properties

Label 1-4000-4000.531-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.997 + 0.0674i$
Analytic cond. $429.859$
Root an. cond. $429.859$
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  + (−0.995 + 0.0941i)3-s + (0.951 + 0.309i)7-s + (0.982 − 0.187i)9-s + (−0.612 + 0.790i)11-s + (0.562 − 0.827i)13-s + (−0.968 − 0.248i)17-s + (−0.0941 + 0.995i)19-s + (−0.975 − 0.218i)21-s + (−0.684 − 0.728i)23-s + (−0.960 + 0.278i)27-s + (−0.750 + 0.661i)29-s + (−0.968 − 0.248i)31-s + (0.535 − 0.844i)33-s + (0.278 − 0.960i)37-s + (−0.481 + 0.876i)39-s + ⋯
L(s)  = 1  + (−0.995 + 0.0941i)3-s + (0.951 + 0.309i)7-s + (0.982 − 0.187i)9-s + (−0.612 + 0.790i)11-s + (0.562 − 0.827i)13-s + (−0.968 − 0.248i)17-s + (−0.0941 + 0.995i)19-s + (−0.975 − 0.218i)21-s + (−0.684 − 0.728i)23-s + (−0.960 + 0.278i)27-s + (−0.750 + 0.661i)29-s + (−0.968 − 0.248i)31-s + (0.535 − 0.844i)33-s + (0.278 − 0.960i)37-s + (−0.481 + 0.876i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.997 + 0.0674i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.997 + 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8631163925 + 0.02916035845i\)
\(L(\frac12)\) \(\approx\) \(0.8631163925 + 0.02916035845i\)
\(L(1)\) \(\approx\) \(0.7123273010 + 0.05425243047i\)
\(L(1)\) \(\approx\) \(0.7123273010 + 0.05425243047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.995 + 0.0941i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.612 + 0.790i)T \)
13 \( 1 + (0.562 - 0.827i)T \)
17 \( 1 + (-0.968 - 0.248i)T \)
19 \( 1 + (-0.0941 + 0.995i)T \)
23 \( 1 + (-0.684 - 0.728i)T \)
29 \( 1 + (-0.750 + 0.661i)T \)
31 \( 1 + (-0.968 - 0.248i)T \)
37 \( 1 + (0.278 - 0.960i)T \)
41 \( 1 + (-0.684 + 0.728i)T \)
43 \( 1 + (-0.987 + 0.156i)T \)
47 \( 1 + (-0.929 - 0.368i)T \)
53 \( 1 + (-0.975 - 0.218i)T \)
59 \( 1 + (-0.338 - 0.940i)T \)
61 \( 1 + (-0.999 - 0.0314i)T \)
67 \( 1 + (-0.661 + 0.750i)T \)
71 \( 1 + (0.368 - 0.929i)T \)
73 \( 1 + (0.904 - 0.425i)T \)
79 \( 1 + (-0.637 + 0.770i)T \)
83 \( 1 + (-0.0941 + 0.995i)T \)
89 \( 1 + (-0.904 + 0.425i)T \)
97 \( 1 + (0.0627 - 0.998i)T \)
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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

−18.27256911834625900918003111757, −17.61539466634048985705004865385, −16.98869429309712508799716354779, −16.40850554294527780363209294923, −15.596622523109157990040670594225, −15.16278409390170745780889181750, −13.98526640547691091114074460673, −13.488906568793861324357347027699, −12.95186625415415290882188389207, −11.82546608019205588843907214844, −11.33788086046014855653607682703, −10.98312898369213735585939256322, −10.25079272071388872068423163075, −9.27249732743623485350696689348, −8.513327063783845446522331071067, −7.74071387085852584124227329661, −7.01019118089208993893730275809, −6.26803872336643270283844285291, −5.586128365749812269189102294866, −4.78723050068681910127579727511, −4.26492783478743503269693885732, −3.32304088360795372246816683867, −1.98326221931325887399332373661, −1.50988891007211405639737219847, −0.37477360644749495462864839531, 0.32460713726578322078820313722, 1.614879280946335875316878864791, 2.00445118703918453551133637311, 3.32180848868063062507203636769, 4.27601858929893254264282869595, 4.90111325064957849897036297431, 5.522805414940834548293334818768, 6.19960303308758543865295509191, 7.065214239748240057863810992205, 7.847709138030347803781182576023, 8.40995608482126717347666729437, 9.47329475814621671402312647827, 10.164857638466809776852654709721, 10.95898418906325154990475983491, 11.20552807210350057306171002101, 12.2273375460004475896460877381, 12.678143612972072130633334746291, 13.33481059781671318428626782489, 14.39439281463278250938046669219, 15.07086702327220726503744862330, 15.57831382702442751994754772884, 16.37746020696165056333443960200, 16.94633001724449457489810846932, 17.88789875596706293659783292304, 18.2543159031702135905386078408

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