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Properties

Label 1-4000-4000.19-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.767 + 0.641i$
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  + (−0.750 + 0.661i)3-s + (−0.951 − 0.309i)7-s + (0.125 − 0.992i)9-s + (−0.940 − 0.338i)11-s + (0.612 + 0.790i)13-s + (−0.637 − 0.770i)17-s + (0.661 − 0.750i)19-s + (0.917 − 0.397i)21-s + (−0.481 + 0.876i)23-s + (0.562 + 0.827i)27-s + (0.218 − 0.975i)29-s + (0.637 + 0.770i)31-s + (0.929 − 0.368i)33-s + (0.827 + 0.562i)37-s + (−0.982 − 0.187i)39-s + ⋯
L(s)  = 1  + (−0.750 + 0.661i)3-s + (−0.951 − 0.309i)7-s + (0.125 − 0.992i)9-s + (−0.940 − 0.338i)11-s + (0.612 + 0.790i)13-s + (−0.637 − 0.770i)17-s + (0.661 − 0.750i)19-s + (0.917 − 0.397i)21-s + (−0.481 + 0.876i)23-s + (0.562 + 0.827i)27-s + (0.218 − 0.975i)29-s + (0.637 + 0.770i)31-s + (0.929 − 0.368i)33-s + (0.827 + 0.562i)37-s + (−0.982 − 0.187i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.767 + 0.641i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.767 + 0.641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7026260081 + 0.2548326252i\)
\(L(\frac12)\) \(\approx\) \(0.7026260081 + 0.2548326252i\)
\(L(1)\) \(\approx\) \(0.6485464157 + 0.08409848847i\)
\(L(1)\) \(\approx\) \(0.6485464157 + 0.08409848847i\)

Euler product

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

−18.120137841813043064134018676956, −17.833550761725308718894617072188, −16.85034226446162357427618771857, −16.147583231582115992163434096848, −15.78221447923903792558216144961, −14.95164087494657063125845020021, −13.93964229801855297094892497882, −13.21962883481412206195852956428, −12.6449807599468623226129847437, −12.38059948345720597050286318930, −11.3519126109068690887824770270, −10.55721946622248953734324240101, −10.23472663236244551502396751678, −9.24021125443545303418153118640, −8.25961765839629464013824240727, −7.75871316085807020019467060292, −6.91592935056706907906627472512, −6.024338692134870061909561341670, −5.85130645495584939206984136748, −4.86872826230496343896960744863, −3.96023408725759211296883039980, −2.924841913042715386951206905066, −2.27525408634517468770244199319, −1.254053554864141994892548777058, −0.30097969511295345233689961621, 0.389228302971258481191676716551, 1.32347947924215888754486631101, 2.78134809300678088361740315644, 3.22754171266906734435981027300, 4.32376839621033423394302497757, 4.730530279152302627455924629314, 5.79016657123090422125078390689, 6.26890345158534669796032918978, 7.005805382172057391577006545978, 7.83299794962095636633880468749, 8.90803378547582627305583990567, 9.56500656533879133139971437369, 9.9757989112216320786193646680, 10.95989053711695998985953087993, 11.33883142701886657162250433110, 12.06952613272329868577745896804, 12.99788519954185264959442615403, 13.56179493265569013395606394689, 14.15830090019640633835230167961, 15.47714901193114454431947465689, 15.7417033048050787817385135411, 16.20155646109744035430824332420, 16.904986718516908415552212559671, 17.759162899758574667991436280047, 18.20330420821088952399964197636

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