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Properties

Label 1-4000-4000.453-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.969 - 0.244i$
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  + (−0.509 − 0.860i)3-s + (0.309 + 0.951i)7-s + (−0.481 + 0.876i)9-s + (−0.562 + 0.827i)11-s + (0.278 + 0.960i)13-s + (0.368 − 0.929i)17-s + (−0.509 + 0.860i)19-s + (0.661 − 0.750i)21-s + (−0.425 + 0.904i)23-s + (0.999 − 0.0314i)27-s + (−0.995 − 0.0941i)29-s + (−0.929 − 0.368i)31-s + (0.998 + 0.0627i)33-s + (0.0314 − 0.999i)37-s + (0.684 − 0.728i)39-s + ⋯
L(s)  = 1  + (−0.509 − 0.860i)3-s + (0.309 + 0.951i)7-s + (−0.481 + 0.876i)9-s + (−0.562 + 0.827i)11-s + (0.278 + 0.960i)13-s + (0.368 − 0.929i)17-s + (−0.509 + 0.860i)19-s + (0.661 − 0.750i)21-s + (−0.425 + 0.904i)23-s + (0.999 − 0.0314i)27-s + (−0.995 − 0.0941i)29-s + (−0.929 − 0.368i)31-s + (0.998 + 0.0627i)33-s + (0.0314 − 0.999i)37-s + (0.684 − 0.728i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.969 - 0.244i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.969 - 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09083941121 + 0.7329524862i\)
\(L(\frac12)\) \(\approx\) \(-0.09083941121 + 0.7329524862i\)
\(L(1)\) \(\approx\) \(0.7714750829 + 0.1250746929i\)
\(L(1)\) \(\approx\) \(0.7714750829 + 0.1250746929i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.509 - 0.860i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.562 + 0.827i)T \)
13 \( 1 + (0.278 + 0.960i)T \)
17 \( 1 + (0.368 - 0.929i)T \)
19 \( 1 + (-0.509 + 0.860i)T \)
23 \( 1 + (-0.425 + 0.904i)T \)
29 \( 1 + (-0.995 - 0.0941i)T \)
31 \( 1 + (-0.929 - 0.368i)T \)
37 \( 1 + (0.0314 - 0.999i)T \)
41 \( 1 + (-0.904 + 0.425i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + (0.844 + 0.535i)T \)
53 \( 1 + (0.750 + 0.661i)T \)
59 \( 1 + (0.790 + 0.612i)T \)
61 \( 1 + (-0.338 - 0.940i)T \)
67 \( 1 + (-0.995 + 0.0941i)T \)
71 \( 1 + (0.844 + 0.535i)T \)
73 \( 1 + (-0.992 - 0.125i)T \)
79 \( 1 + (0.968 + 0.248i)T \)
83 \( 1 + (0.860 + 0.509i)T \)
89 \( 1 + (-0.125 + 0.992i)T \)
97 \( 1 + (-0.770 + 0.637i)T \)
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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

−17.84447262306883306122799469289, −17.06601557399574493279764029437, −16.68534973359002579632744690086, −16.02548496670917698658405212767, −15.114944222107741631017982157135, −14.85116174023210987677796126713, −13.798743186448247492608442098495, −13.223671386313397596411526054613, −12.45636474468509466504675335685, −11.55203232725655152744097418732, −10.73781756911175803431438687456, −10.56960611307415024431872191148, −9.93702162854700365820956688708, −8.71657962114076830857125153632, −8.41556602485145944280624603576, −7.4146473443427167967843249995, −6.587439371575908239896449097415, −5.7237210481570866592649760165, −5.23602323548243604479600403563, −4.33576898920474486249732511907, −3.65206059601728781605814934308, −3.06273632283049928496946450071, −1.80527437566016513177368744792, −0.59250208157942812497104056739, −0.17951930794714042194711956071, 1.20088938094805734897596345872, 2.01122469380904024239336572201, 2.39720536767830621953827848705, 3.638458421679435681775283720505, 4.639846574437014134229715621870, 5.435679640859976581701276444725, 5.8759077601410642357011692355, 6.765832958120761349358228473692, 7.563856453968619730715305515147, 7.95328400906195627407123034635, 9.01716947881639460257488308986, 9.54268446576032108303068510556, 10.55595878283257914530391818130, 11.35785602809145979136406524114, 11.85179207694897345865430944411, 12.44138646915241319246437673979, 13.087272606701017396936829927118, 13.84703735268035466143829196040, 14.56802873017562472248245687812, 15.22592281283361580335298260863, 16.15849566961024072218561103717, 16.60747530462117431255990591083, 17.48504041287697403990997212682, 18.19883308972914693687080953994, 18.505876423040851628884480825523

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