↑

Properties

Label 1-4000-4000.1413-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.0800 + 0.996i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0800 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0800 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.0800 + 0.996i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.0800 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.055336400 + 0.9740069309i\)
\(L(\frac12)\) \(\approx\) \(1.055336400 + 0.9740069309i\)
\(L(1)\) \(\approx\) \(0.8731640656 + 0.01114953271i\)
\(L(1)\) \(\approx\) \(0.8731640656 + 0.01114953271i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−17.73445630144833947866488554914, −17.46920552922389052998141648017, −16.81015244801432658021437328697, −15.98001985152859101218623437433, −15.598670018764272018077160310849, −14.96118469325633896405967849197, −13.91319758157792660072244880851, −13.60080221886328622730072823089, −12.635291747379320114328638760422, −11.78765655748101925511733782867, −11.22520738393733695159648950458, −10.521630786145502209570372677334, −9.738522665919884452987405438998, −9.40737388513764588919420793012, −8.54427345130213002284727080352, −7.68726738404189814387062271281, −6.644165903355432113858588651980, −6.24131930588787575541097054373, −5.323919939758598119426199967346, −4.514861620175509427604029642768, −3.87905496648563545067485383513, −3.14734169230277820249476818315, −2.43299102472116936274861133796, −0.82244811741486935086666492543, −0.34998650869158668306816294818, 0.77533045016737030176337435875, 1.88008292410193468189535385893, 2.19647298109282876689411349084, 3.36212347187891536032297097990, 4.16572621568644211145290714688, 5.09405697506596355208659409879, 6.1115394863659684414619132358, 6.511155524761967889721872170776, 6.97105922942285465729748567554, 8.04325239748507131451503012091, 8.73628493703632533702113184070, 9.29006239298484199483769336288, 10.249625251446818123479174707820, 10.96819796018462052702526652336, 11.93071045928954944906367995579, 12.3887634028055233937184613843, 12.65294205447055591302033788695, 13.842897433303228551455583927679, 14.16581805651139368331161434570, 14.98682769173533584921113250135, 15.88633970109784660626613976761, 16.58955430554027989580377533768, 17.117290691481567911932657223365, 17.83490278924029282495864725395, 18.598964189632721703443143084263

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