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Properties

Label 1-4024-4024.373-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.664 - 0.746i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (0.739 + 0.673i)3-s + (−0.580 − 0.814i)5-s + (0.992 + 0.124i)7-s + (0.0937 + 0.995i)9-s + (0.630 − 0.776i)11-s + (−0.143 + 0.989i)13-s + (0.118 − 0.992i)15-s + (−0.485 − 0.874i)17-s + (−0.731 + 0.682i)19-s + (0.649 + 0.760i)21-s + (−0.441 − 0.897i)23-s + (−0.325 + 0.945i)25-s + (−0.600 + 0.799i)27-s + (−0.920 − 0.389i)29-s + (−0.349 − 0.937i)31-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)3-s + (−0.580 − 0.814i)5-s + (0.992 + 0.124i)7-s + (0.0937 + 0.995i)9-s + (0.630 − 0.776i)11-s + (−0.143 + 0.989i)13-s + (0.118 − 0.992i)15-s + (−0.485 − 0.874i)17-s + (−0.731 + 0.682i)19-s + (0.649 + 0.760i)21-s + (−0.441 − 0.897i)23-s + (−0.325 + 0.945i)25-s + (−0.600 + 0.799i)27-s + (−0.920 − 0.389i)29-s + (−0.349 − 0.937i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.664 - 0.746i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.664 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2724480882 - 0.6073810803i\)
\(L(\frac12)\) \(\approx\) \(0.2724480882 - 0.6073810803i\)
\(L(1)\) \(\approx\) \(1.080072909 + 0.01539270520i\)
\(L(1)\) \(\approx\) \(1.080072909 + 0.01539270520i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (-0.580 - 0.814i)T \)
7 \( 1 + (0.992 + 0.124i)T \)
11 \( 1 + (0.630 - 0.776i)T \)
13 \( 1 + (-0.143 + 0.989i)T \)
17 \( 1 + (-0.485 - 0.874i)T \)
19 \( 1 + (-0.731 + 0.682i)T \)
23 \( 1 + (-0.441 - 0.897i)T \)
29 \( 1 + (-0.920 - 0.389i)T \)
31 \( 1 + (-0.349 - 0.937i)T \)
37 \( 1 + (-0.496 - 0.868i)T \)
41 \( 1 + (-0.883 - 0.468i)T \)
43 \( 1 + (-0.998 + 0.0500i)T \)
47 \( 1 + (0.0437 + 0.999i)T \)
53 \( 1 + (-0.731 - 0.682i)T \)
59 \( 1 + (-0.947 + 0.319i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (-0.118 - 0.992i)T \)
71 \( 1 + (0.429 + 0.902i)T \)
73 \( 1 + (-0.155 - 0.987i)T \)
79 \( 1 + (0.659 - 0.752i)T \)
83 \( 1 + (-0.955 + 0.295i)T \)
89 \( 1 + (-0.999 + 0.0375i)T \)
97 \( 1 + (-0.507 - 0.861i)T \)
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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

−18.64222914252101381911428086718, −18.11817868209834299646370006862, −17.496848113751300800597119895677, −17.02796274243188285062194273207, −15.53624772353266829655425538404, −15.07413925568167523576551577386, −14.85958616927790537810546074787, −14.00508242585194847715208206867, −13.35705846392907508457363583778, −12.52005269134933241437374097441, −11.9447402897794998401984409614, −11.188729757450676661510495254371, −10.527286872506768601843589973246, −9.738209509751830690624032271024, −8.71026532607415415737497498587, −8.23370728647105477605422715777, −7.53306216462650988731501554415, −6.966647860490515886036651496820, −6.351903075938732739022602075703, −5.23250719610213920109014725725, −4.29175987598939827031347253981, −3.61128905116021293488755378155, −2.88632921870837252113852906452, −1.879553142068577855148280968537, −1.45140530128187374840993136398, 0.14694086860606259428247056632, 1.63341647814257380726454770611, 2.07693142452036660579044124772, 3.29055345618709993341303331324, 4.13739486333594126242312046267, 4.44850363165628528225901850454, 5.22471069969471691385250119016, 6.14212478600347535554981394130, 7.262082394358990056096634492246, 7.97771933172144948189868810872, 8.566851369835463144608963851453, 9.06774327739023784927304686549, 9.656175832266703665896316109251, 10.811147635720753207763745477968, 11.301264295398524335472631513104, 11.927397391540481249208586904861, 12.74191212399192914540957342038, 13.76983635510117420001336942987, 14.08729778505939532913325350795, 14.88698936505297751874373903443, 15.399902438027936599804732056531, 16.35485981923316482647649124386, 16.62662950494281235689967809385, 17.26973336560608805258987672261, 18.52772675126877663374619285518

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