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Properties

Label 1-4024-4024.1229-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.372 + 0.927i$
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  + (0.610 − 0.791i)3-s + (−0.731 + 0.682i)5-s + (−0.180 − 0.983i)7-s + (−0.253 − 0.967i)9-s + (−0.992 + 0.124i)11-s + (−0.429 − 0.902i)13-s + (0.0937 + 0.995i)15-s + (−0.686 − 0.726i)17-s + (0.463 + 0.886i)19-s + (−0.889 − 0.457i)21-s + (−0.993 − 0.112i)23-s + (0.0687 − 0.997i)25-s + (−0.920 − 0.389i)27-s + (−0.780 − 0.625i)29-s + (−0.277 + 0.960i)31-s + ⋯
L(s)  = 1  + (0.610 − 0.791i)3-s + (−0.731 + 0.682i)5-s + (−0.180 − 0.983i)7-s + (−0.253 − 0.967i)9-s + (−0.992 + 0.124i)11-s + (−0.429 − 0.902i)13-s + (0.0937 + 0.995i)15-s + (−0.686 − 0.726i)17-s + (0.463 + 0.886i)19-s + (−0.889 − 0.457i)21-s + (−0.993 − 0.112i)23-s + (0.0687 − 0.997i)25-s + (−0.920 − 0.389i)27-s + (−0.780 − 0.625i)29-s + (−0.277 + 0.960i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.372 + 0.927i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.372 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2104046408 + 0.1422208181i\)
\(L(\frac12)\) \(\approx\) \(0.2104046408 + 0.1422208181i\)
\(L(1)\) \(\approx\) \(0.7609918354 - 0.2776338114i\)
\(L(1)\) \(\approx\) \(0.7609918354 - 0.2776338114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.610 - 0.791i)T \)
5 \( 1 + (-0.731 + 0.682i)T \)
7 \( 1 + (-0.180 - 0.983i)T \)
11 \( 1 + (-0.992 + 0.124i)T \)
13 \( 1 + (-0.429 - 0.902i)T \)
17 \( 1 + (-0.686 - 0.726i)T \)
19 \( 1 + (0.463 + 0.886i)T \)
23 \( 1 + (-0.993 - 0.112i)T \)
29 \( 1 + (-0.780 - 0.625i)T \)
31 \( 1 + (-0.277 + 0.960i)T \)
37 \( 1 + (0.549 + 0.835i)T \)
41 \( 1 + (0.852 - 0.523i)T \)
43 \( 1 + (-0.764 - 0.644i)T \)
47 \( 1 + (-0.817 - 0.575i)T \)
53 \( 1 + (0.463 - 0.886i)T \)
59 \( 1 + (0.155 + 0.987i)T \)
61 \( 1 + (0.507 + 0.861i)T \)
67 \( 1 + (-0.0937 + 0.995i)T \)
71 \( 1 + (-0.998 + 0.0625i)T \)
73 \( 1 + (0.580 - 0.814i)T \)
79 \( 1 + (0.795 - 0.605i)T \)
83 \( 1 + (0.485 + 0.874i)T \)
89 \( 1 + (0.864 + 0.501i)T \)
97 \( 1 + (-0.395 - 0.918i)T \)
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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

−18.50872215027490534822303187571, −17.66690785043795080923256096931, −16.669239928390106493644430829528, −16.10703553535549303538322963246, −15.726840741410813324618190637186, −15.02250528604543623941518221111, −14.53588303149008460525028129178, −13.43861110103111553792036674650, −12.961588712789465261962972938298, −12.20851226157250414417701380999, −11.25649208495216718848028405443, −11.00751978472863595753366233232, −9.6715849534852401043003416876, −9.39087989947788335226612127571, −8.654752631375449236656057153, −8.02028429189466702044063618329, −7.447906662790498782432475166063, −6.23784486178677159258352213306, −5.37462371952411464813956698185, −4.76607413855412609526069134032, −4.11933125464356209549144486254, −3.301787552673299101056993188367, −2.4330567599088244799312838529, −1.82533669868439641085495738854, −0.080123741680526166122810998452, 0.778212881837776190670704887599, 2.02730225949887547462472661522, 2.75837857336954502710699161788, 3.45198489966537611744994010663, 4.08130385360701903675937692126, 5.15195813047372693195032820000, 6.11270091133412859942333774129, 6.972840160200536346823805997599, 7.47345019707576468314451732829, 7.91134869568052185547539234670, 8.58027217265507090367955770748, 9.84019930757829183913480041965, 10.206850320175561197948787599420, 11.04265056187231336950192755631, 11.86916729711505469445889530871, 12.43722468912691007068752442413, 13.44683235733437677447389837405, 13.55104064825428082747394834044, 14.64601509358809564206611469294, 14.95356582899550417969227888320, 15.90173553158304413180181434880, 16.387462799798364439612377073385, 17.55375416399099692871285907576, 18.085987589479977009186318439771, 18.48929073594352500525132367401

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