↑

Properties

Label 1-4005-4005.1138-r0-0-0
Degree $1$
Conductor $4005$
Sign $-0.956 - 0.290i$
Analytic cond. $18.5991$
Root an. cond. $18.5991$
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  + (−0.945 + 0.327i)2-s + (0.786 − 0.618i)4-s + (0.304 − 0.952i)7-s + (−0.540 + 0.841i)8-s + (−0.0475 − 0.998i)11-s + (−0.919 − 0.393i)13-s + (0.0237 + 0.999i)14-s + (0.235 − 0.971i)16-s + (0.654 + 0.755i)17-s + (0.877 − 0.479i)19-s + (0.371 + 0.928i)22-s + (0.853 + 0.520i)23-s + (0.997 + 0.0713i)26-s + (−0.349 − 0.936i)28-s + (0.304 − 0.952i)29-s + ⋯
L(s)  = 1  + (−0.945 + 0.327i)2-s + (0.786 − 0.618i)4-s + (0.304 − 0.952i)7-s + (−0.540 + 0.841i)8-s + (−0.0475 − 0.998i)11-s + (−0.919 − 0.393i)13-s + (0.0237 + 0.999i)14-s + (0.235 − 0.971i)16-s + (0.654 + 0.755i)17-s + (0.877 − 0.479i)19-s + (0.371 + 0.928i)22-s + (0.853 + 0.520i)23-s + (0.997 + 0.0713i)26-s + (−0.349 − 0.936i)28-s + (0.304 − 0.952i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
Sign: $-0.956 - 0.290i$
Analytic conductor: \(18.5991\)
Root analytic conductor: \(18.5991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4005} (1138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4005,\ (0:\ ),\ -0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07217428832 - 0.4865030844i\)
\(L(\frac12)\) \(\approx\) \(0.07217428832 - 0.4865030844i\)
\(L(1)\) \(\approx\) \(0.6325423265 - 0.1214138777i\)
\(L(1)\) \(\approx\) \(0.6325423265 - 0.1214138777i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.945 + 0.327i)T \)
7 \( 1 + (0.304 - 0.952i)T \)
11 \( 1 + (-0.0475 - 0.998i)T \)
13 \( 1 + (-0.919 - 0.393i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
19 \( 1 + (0.877 - 0.479i)T \)
23 \( 1 + (0.853 + 0.520i)T \)
29 \( 1 + (0.304 - 0.952i)T \)
31 \( 1 + (-0.999 + 0.0237i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.992 - 0.118i)T \)
43 \( 1 + (-0.739 - 0.672i)T \)
47 \( 1 + (-0.928 - 0.371i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (0.118 - 0.992i)T \)
61 \( 1 + (-0.165 + 0.986i)T \)
67 \( 1 + (-0.371 - 0.928i)T \)
71 \( 1 + (-0.540 - 0.841i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (-0.971 - 0.235i)T \)
83 \( 1 + (-0.560 + 0.828i)T \)
97 \( 1 + (0.458 + 0.888i)T \)
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

−18.558508532733900713954345184505, −18.28810542529355651797193496637, −17.58881910931550155001395730592, −16.795089510196298572730452198015, −16.26558378182511629772810333017, −15.49128928809985567836624971578, −14.76512699145071612259977243867, −14.305601393154591690114428374961, −12.92984851328192053624115996829, −12.48121802067542120762232982139, −11.69986506959752787378526177164, −11.46606362172145298258388379190, −10.15680215252365114166651619949, −9.90781895649141017776912264938, −9.07418501207600611695428128077, −8.55074565605078750893970908010, −7.55521945425834248552776614447, −7.17747040362377235666183443679, −6.340064668057262330783905017930, −5.17954244197829167608835424124, −4.78894195254913515736629763644, −3.366825952944794273746740090, −2.83003588942517730126713793341, −1.91131574701610426829996124422, −1.34222867613584502234683591435, 0.202367073725999280050622574329, 1.075188693541888566618679305979, 1.87831769733919730193968159632, 3.03022301358380188787439375954, 3.62240376683407189769499061811, 4.94893180574761242804775893018, 5.46331952111260540982760635561, 6.34465231795412247833320821287, 7.26564426736368155104213761170, 7.580843069308514120118518817808, 8.38817134435220928783844894188, 9.07352507253919269503533296495, 9.98456030350786508700276916311, 10.3698324978918360708429670026, 11.19154976959257228519553685775, 11.673865510276041196850048948297, 12.653269572859715983889169552172, 13.60222994688433206595371610774, 14.15248337940970528631062981887, 14.926705409154564059206890966389, 15.51974095291244561352969060194, 16.37604240868893753349488204471, 16.98133441472462430897668858248, 17.297635910527919758851985163629, 18.11649835802460102965156934974

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