Properties

Label 1-4024-4024.85-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.644 - 0.764i$
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  + (0.986 − 0.161i)3-s + (0.965 − 0.259i)5-s + (−0.817 + 0.575i)7-s + (0.947 − 0.319i)9-s + (−0.0437 − 0.999i)11-s + (−0.384 + 0.923i)13-s + (0.910 − 0.412i)15-s + (0.877 − 0.479i)17-s + (0.858 + 0.512i)19-s + (−0.713 + 0.700i)21-s + (−0.418 − 0.908i)23-s + (0.864 − 0.501i)25-s + (0.883 − 0.468i)27-s + (−0.852 − 0.523i)29-s + (0.289 + 0.957i)31-s + ⋯
L(s)  = 1  + (0.986 − 0.161i)3-s + (0.965 − 0.259i)5-s + (−0.817 + 0.575i)7-s + (0.947 − 0.319i)9-s + (−0.0437 − 0.999i)11-s + (−0.384 + 0.923i)13-s + (0.910 − 0.412i)15-s + (0.877 − 0.479i)17-s + (0.858 + 0.512i)19-s + (−0.713 + 0.700i)21-s + (−0.418 − 0.908i)23-s + (0.864 − 0.501i)25-s + (0.883 − 0.468i)27-s + (−0.852 − 0.523i)29-s + (0.289 + 0.957i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.644 - 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.681439842 - 1.246434602i\)
\(L(\frac12)\) \(\approx\) \(2.681439842 - 1.246434602i\)
\(L(1)\) \(\approx\) \(1.653372195 - 0.2705101542i\)
\(L(1)\) \(\approx\) \(1.653372195 - 0.2705101542i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.986 - 0.161i)T \)
5 \( 1 + (0.965 - 0.259i)T \)
7 \( 1 + (-0.817 + 0.575i)T \)
11 \( 1 + (-0.0437 - 0.999i)T \)
13 \( 1 + (-0.384 + 0.923i)T \)
17 \( 1 + (0.877 - 0.479i)T \)
19 \( 1 + (0.858 + 0.512i)T \)
23 \( 1 + (-0.418 - 0.908i)T \)
29 \( 1 + (-0.852 - 0.523i)T \)
31 \( 1 + (0.289 + 0.957i)T \)
37 \( 1 + (-0.992 + 0.124i)T \)
41 \( 1 + (-0.485 - 0.874i)T \)
43 \( 1 + (-0.0687 - 0.997i)T \)
47 \( 1 + (-0.253 - 0.967i)T \)
53 \( 1 + (0.858 - 0.512i)T \)
59 \( 1 + (0.943 + 0.331i)T \)
61 \( 1 + (0.205 - 0.978i)T \)
67 \( 1 + (-0.910 - 0.412i)T \)
71 \( 1 + (-0.722 - 0.691i)T \)
73 \( 1 + (0.0187 + 0.999i)T \)
79 \( 1 + (0.920 + 0.389i)T \)
83 \( 1 + (0.915 - 0.401i)T \)
89 \( 1 + (0.429 + 0.902i)T \)
97 \( 1 + (0.968 + 0.247i)T \)
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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

−18.57463083768481305114657245341, −17.86711766917661340134309417675, −17.31715205611433012170422920272, −16.48170832564891537904953365959, −15.78339288596759863120484370454, −14.92480189726072690541667641209, −14.62852396788138103631293012507, −13.68034634970618504025518619494, −13.20430551932726401955792589099, −12.766777880979065511705084675383, −11.821387340840916696989726022376, −10.52742156636282699417492836842, −10.16542095404262180410836949841, −9.56403723005272433731996928307, −9.18936837905831535803965498922, −7.91595559265753578668438403448, −7.45958371493350826142172805653, −6.81317744369066528334884154356, −5.82986640084071372330778386769, −5.1286904090410444903095269311, −4.16839529748661601691832181563, −3.26608281042459482033152638051, −2.86395567597970760783072306032, −1.88526450800533310871107494609, −1.140565144581060595141760464703, 0.72495754309366864904191917822, 1.832070601628137230729039329843, 2.37244163488848858255385678880, 3.26850856579991328848993610359, 3.76058024561253559695721768328, 5.065333990366660705749101095019, 5.624914038178980005788646040096, 6.53173776847980280276591072994, 7.05255949742632670832224309394, 8.09603733807432570763990364691, 8.84786294970389016935107287012, 9.19945836586442346890677821648, 10.01564904900886548262088633454, 10.38228920813357155151321233052, 11.902374087422601497180435740576, 12.17118887337628093936483635557, 13.10345799353904672202149669481, 13.76570111503396869566807866418, 14.07673568398959484779624001929, 14.793435547723565952831746190707, 15.74181801001068782938465225373, 16.40591351007705909193921231581, 16.75120790472932379735379020388, 17.919268967089125245341850151721, 18.6294165533537685192721790713

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