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Properties

Label 1-4029-4029.1142-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.00705 + 0.999i$
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  + (0.761 − 0.647i)2-s + (0.160 − 0.987i)4-s + (0.0704 − 0.997i)5-s + (−0.993 + 0.110i)7-s + (−0.517 − 0.855i)8-s + (−0.592 − 0.805i)10-s + (−0.559 − 0.828i)11-s + (−0.774 − 0.632i)13-s + (−0.685 + 0.728i)14-s + (−0.948 − 0.316i)16-s + (0.990 − 0.140i)19-s + (−0.973 − 0.229i)20-s + (−0.963 − 0.268i)22-s + (0.130 − 0.991i)23-s + (−0.990 − 0.140i)25-s + (−0.999 + 0.0201i)26-s + ⋯
L(s)  = 1  + (0.761 − 0.647i)2-s + (0.160 − 0.987i)4-s + (0.0704 − 0.997i)5-s + (−0.993 + 0.110i)7-s + (−0.517 − 0.855i)8-s + (−0.592 − 0.805i)10-s + (−0.559 − 0.828i)11-s + (−0.774 − 0.632i)13-s + (−0.685 + 0.728i)14-s + (−0.948 − 0.316i)16-s + (0.990 − 0.140i)19-s + (−0.973 − 0.229i)20-s + (−0.963 − 0.268i)22-s + (0.130 − 0.991i)23-s + (−0.990 − 0.140i)25-s + (−0.999 + 0.0201i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00705 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00705 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.00705 + 0.999i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.00705 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8222108879 - 0.8280302035i\)
\(L(\frac12)\) \(\approx\) \(-0.8222108879 - 0.8280302035i\)
\(L(1)\) \(\approx\) \(0.7199681566 - 0.9132667413i\)
\(L(1)\) \(\approx\) \(0.7199681566 - 0.9132667413i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.761 - 0.647i)T \)
5 \( 1 + (0.0704 - 0.997i)T \)
7 \( 1 + (-0.993 + 0.110i)T \)
11 \( 1 + (-0.559 - 0.828i)T \)
13 \( 1 + (-0.774 - 0.632i)T \)
19 \( 1 + (0.990 - 0.140i)T \)
23 \( 1 + (0.130 - 0.991i)T \)
29 \( 1 + (-0.170 - 0.985i)T \)
31 \( 1 + (-0.0503 - 0.998i)T \)
37 \( 1 + (-0.999 + 0.0100i)T \)
41 \( 1 + (0.326 - 0.945i)T \)
43 \( 1 + (-0.834 + 0.551i)T \)
47 \( 1 + (-0.391 + 0.919i)T \)
53 \( 1 + (0.0201 + 0.999i)T \)
59 \( 1 + (0.811 + 0.584i)T \)
61 \( 1 + (0.963 - 0.268i)T \)
67 \( 1 + (-0.568 - 0.822i)T \)
71 \( 1 + (0.805 + 0.592i)T \)
73 \( 1 + (0.957 - 0.287i)T \)
83 \( 1 + (0.811 - 0.584i)T \)
89 \( 1 + (-0.239 - 0.970i)T \)
97 \( 1 + (-0.871 - 0.491i)T \)
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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

−18.98154133656293919990027475776, −17.976685661084551233655646197649, −17.75722583117803660463795377561, −16.70569280936916637856999337677, −16.162956380677397148215300549540, −15.484838769954631111353531049509, −14.91477141167196245657592126595, −14.246435885255488952978623259867, −13.62632140939627456910434422152, −12.98330676883959836126033500956, −12.20385079217449841204804195383, −11.68432653167773711570393815987, −10.75359660491808498036831462745, −9.88128914662444594644827734496, −9.46095261173219898248413950331, −8.327995030855662529453715776330, −7.396455512807655952194366137808, −6.93484950318297836351278925499, −6.622083419036613857687815484896, −5.39020693734127479783769191533, −5.101272883598149025698706249, −3.833739567812706820432861927692, −3.36266942196613302313263838545, −2.63105940258165213401250204176, −1.806042921598283785411082189328, 0.277655909308162176290413800777, 0.88688490434564986578546264805, 2.16300598987174737086945804307, 2.82489275272935781379086808758, 3.54852480886992875364195624290, 4.393399003941820051829730753680, 5.22157587297434421380495632005, 5.687648448976083922057944063257, 6.394497991573868301142947727382, 7.414683396514639931998352616959, 8.284576259429235617508861179275, 9.18878685428235040281033306809, 9.74075468894587528828171192745, 10.339980512640436401199049021638, 11.20016775991167811130186010411, 12.04702226650159086883601808142, 12.48175129179148676682197858432, 13.209130606971211532273648132942, 13.53663517239525378097405298811, 14.37203739020319134475373459915, 15.36216646290796226512609536136, 15.79921965281277182917909580065, 16.455664183097511446008156397859, 17.1552229842657165468997876375, 18.145904505480039184543872209170

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