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Properties

Label 1-4000-4000.1477-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.455 - 0.890i$
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  + (0.960 + 0.278i)3-s + (0.809 + 0.587i)7-s + (0.844 + 0.535i)9-s + (0.397 + 0.917i)11-s + (−0.975 + 0.218i)13-s + (−0.684 − 0.728i)17-s + (−0.960 + 0.278i)19-s + (0.612 + 0.790i)21-s + (0.637 − 0.770i)23-s + (0.661 + 0.750i)27-s + (−0.827 − 0.562i)29-s + (0.728 − 0.684i)31-s + (0.125 + 0.992i)33-s + (−0.750 − 0.661i)37-s + (−0.998 − 0.0627i)39-s + ⋯
L(s)  = 1  + (0.960 + 0.278i)3-s + (0.809 + 0.587i)7-s + (0.844 + 0.535i)9-s + (0.397 + 0.917i)11-s + (−0.975 + 0.218i)13-s + (−0.684 − 0.728i)17-s + (−0.960 + 0.278i)19-s + (0.612 + 0.790i)21-s + (0.637 − 0.770i)23-s + (0.661 + 0.750i)27-s + (−0.827 − 0.562i)29-s + (0.728 − 0.684i)31-s + (0.125 + 0.992i)33-s + (−0.750 − 0.661i)37-s + (−0.998 − 0.0627i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.455 - 0.890i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.455 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.966614458 - 1.203019309i\)
\(L(\frac12)\) \(\approx\) \(1.966614458 - 1.203019309i\)
\(L(1)\) \(\approx\) \(1.420815579 + 0.1647426824i\)
\(L(1)\) \(\approx\) \(1.420815579 + 0.1647426824i\)

Euler product

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

−18.5929767574145741302447517363, −17.64338458024884623971118466769, −17.19181353633094200434090847420, −16.5340481698523844830905016163, −15.30195917694676442232957150590, −15.07283892068708984632868789440, −14.347814996819829109342777039125, −13.59370942081932226111300254779, −13.26254855396524260116150361173, −12.30542726481384778302739688445, −11.61946031504418004887442560290, −10.69130992151229019018705235078, −10.25647150479378813682007378996, −9.17146427150954328461955297668, −8.603756504665425170842997964524, −8.13185299858705533896705892801, −7.127758456130027877539739508609, −6.854787701265062079489054087272, −5.689793083702119489041467438476, −4.77484226452974711452752511360, −4.053786905459908361615485618270, −3.34639228119666735523647567020, −2.472840502840999666504902684902, −1.644651945654117750958946008390, −0.95687459880796138577820747756, 0.27299030703107310261120605714, 1.82312315271561878082854309119, 2.09252274077953474211990970949, 2.876367949988248806233837278406, 4.00250658786915836859159059998, 4.66099893498194561225313177666, 5.06477737870604660316115300209, 6.35430159583362254620791161979, 7.14364215646170530501383572766, 7.73828377922413114449954706056, 8.57287170907501024711538227399, 9.08712917696063498217092386223, 9.76407083292528877889416206108, 10.45396140194642591113372702593, 11.35445402566414932958114029865, 12.07295878882636487368017911085, 12.7391764472735402332157719485, 13.49952274193852716111247845937, 14.33665227697579849486653652388, 14.84620759280802859743246243563, 15.20618255946109118644435704853, 15.94246548703083783418815709292, 17.07625226033281684014683949036, 17.30814698091206797334853784545, 18.47317860469260048125106436209

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