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Properties

Label 1-4029-4029.1232-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.218 + 0.975i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (−0.160 + 0.987i)2-s + (−0.948 − 0.316i)4-s + (−0.990 − 0.140i)5-s + (0.975 − 0.219i)7-s + (0.464 − 0.885i)8-s + (0.297 − 0.954i)10-s + (−0.373 + 0.927i)11-s + (0.200 + 0.979i)13-s + (0.0603 + 0.998i)14-s + (0.799 + 0.600i)16-s + (0.960 − 0.278i)19-s + (0.894 + 0.446i)20-s + (−0.855 − 0.517i)22-s + (−0.965 − 0.258i)23-s + (0.960 + 0.278i)25-s + (−0.999 + 0.0402i)26-s + ⋯
L(s)  = 1  + (−0.160 + 0.987i)2-s + (−0.948 − 0.316i)4-s + (−0.990 − 0.140i)5-s + (0.975 − 0.219i)7-s + (0.464 − 0.885i)8-s + (0.297 − 0.954i)10-s + (−0.373 + 0.927i)11-s + (0.200 + 0.979i)13-s + (0.0603 + 0.998i)14-s + (0.799 + 0.600i)16-s + (0.960 − 0.278i)19-s + (0.894 + 0.446i)20-s + (−0.855 − 0.517i)22-s + (−0.965 − 0.258i)23-s + (0.960 + 0.278i)25-s + (−0.999 + 0.0402i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.218 + 0.975i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.218 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8144288784 + 1.016606522i\)
\(L(\frac12)\) \(\approx\) \(0.8144288784 + 1.016606522i\)
\(L(1)\) \(\approx\) \(0.7710947237 + 0.4393233127i\)
\(L(1)\) \(\approx\) \(0.7710947237 + 0.4393233127i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.160 + 0.987i)T \)
5 \( 1 + (-0.990 - 0.140i)T \)
7 \( 1 + (0.975 - 0.219i)T \)
11 \( 1 + (-0.373 + 0.927i)T \)
13 \( 1 + (0.200 + 0.979i)T \)
19 \( 1 + (0.960 - 0.278i)T \)
23 \( 1 + (-0.965 - 0.258i)T \)
29 \( 1 + (0.941 - 0.335i)T \)
31 \( 1 + (0.994 - 0.100i)T \)
37 \( 1 + (0.999 - 0.0201i)T \)
41 \( 1 + (0.787 + 0.616i)T \)
43 \( 1 + (-0.391 + 0.919i)T \)
47 \( 1 + (-0.692 - 0.721i)T \)
53 \( 1 + (-0.999 + 0.0402i)T \)
59 \( 1 + (0.316 + 0.948i)T \)
61 \( 1 + (0.855 - 0.517i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (-0.297 - 0.954i)T \)
73 \( 1 + (-0.834 + 0.551i)T \)
83 \( 1 + (-0.316 + 0.948i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.517 - 0.855i)T \)
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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

−18.33251845392627112468962859632, −17.86513733509074286734034238797, −17.1897162965754943181192615910, −16.00211335719994138559806594010, −15.80054405481817060305439527601, −14.639538508209108809898925038180, −14.18462917984196462969067872895, −13.40469726920704314252064396567, −12.62040826060034913205146839716, −11.767318943054366712833801315069, −11.60518890089663807525556164742, −10.69115161579448267722217505677, −10.32714001470023878268824112615, −9.26370425908017206723989736933, −8.35401555006867265546076679145, −8.056683430295352671954538393475, −7.52460366468415366027689600515, −6.12194315107317736519945943299, −5.27513644424169882335929923423, −4.6605211353636708859903106699, −3.77186116084467792168163563203, −3.13896344357204511918237256740, −2.462962120975178167054823104174, −1.27290086790990513282709805666, −0.59818893257685572243555421001, 0.8063781492041521913780964026, 1.67152367355332791100804456521, 2.91052957977137073856636264640, 4.20421434029206382068033855101, 4.42682411444307720464754428493, 5.05732440281095676442908837499, 6.11270916819524617452272313091, 6.92213677851667430229854253629, 7.54955703082582931103166644540, 8.09356097764313551088222928666, 8.61728414964452883195968287986, 9.62363177935560952479964189567, 10.14818064503335679790719344751, 11.25319831413129729865973969580, 11.7412711042904580027657469512, 12.52544736629192117953711894097, 13.417823383609652947901862691038, 14.11267736065958517602233147734, 14.71488170115968378372875433991, 15.27707619558840953929779043299, 16.087120038635095602438504382912, 16.35680332765653551996363831340, 17.34294772394218319146095408698, 17.9645114372173979920919222760, 18.3643453135214559864490004958

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