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Properties

Label 1-4029-4029.62-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.719 - 0.694i$
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  + (−0.297 − 0.954i)2-s + (−0.822 + 0.568i)4-s + (−0.728 + 0.685i)5-s + (−0.871 − 0.491i)7-s + (0.787 + 0.616i)8-s + (0.871 + 0.491i)10-s + (0.685 − 0.728i)11-s + (0.822 − 0.568i)13-s + (−0.209 + 0.977i)14-s + (0.354 − 0.935i)16-s + (−0.0603 − 0.998i)19-s + (0.209 − 0.977i)20-s + (−0.899 − 0.437i)22-s + (0.923 + 0.382i)23-s + (0.0603 − 0.998i)25-s + (−0.787 − 0.616i)26-s + ⋯
L(s)  = 1  + (−0.297 − 0.954i)2-s + (−0.822 + 0.568i)4-s + (−0.728 + 0.685i)5-s + (−0.871 − 0.491i)7-s + (0.787 + 0.616i)8-s + (0.871 + 0.491i)10-s + (0.685 − 0.728i)11-s + (0.822 − 0.568i)13-s + (−0.209 + 0.977i)14-s + (0.354 − 0.935i)16-s + (−0.0603 − 0.998i)19-s + (0.209 − 0.977i)20-s + (−0.899 − 0.437i)22-s + (0.923 + 0.382i)23-s + (0.0603 − 0.998i)25-s + (−0.787 − 0.616i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.719 - 0.694i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.719 - 0.694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3783526101 - 0.9358783131i\)
\(L(\frac12)\) \(\approx\) \(0.3783526101 - 0.9358783131i\)
\(L(1)\) \(\approx\) \(0.6537799500 - 0.3907354926i\)
\(L(1)\) \(\approx\) \(0.6537799500 - 0.3907354926i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.297 - 0.954i)T \)
5 \( 1 + (-0.728 + 0.685i)T \)
7 \( 1 + (-0.871 - 0.491i)T \)
11 \( 1 + (0.685 - 0.728i)T \)
13 \( 1 + (0.822 - 0.568i)T \)
19 \( 1 + (-0.0603 - 0.998i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.592 - 0.805i)T \)
31 \( 1 + (0.995 + 0.0904i)T \)
37 \( 1 + (0.945 + 0.326i)T \)
41 \( 1 + (0.999 + 0.0302i)T \)
43 \( 1 + (-0.911 - 0.410i)T \)
47 \( 1 + (-0.663 - 0.748i)T \)
53 \( 1 + (0.616 - 0.787i)T \)
59 \( 1 + (-0.180 - 0.983i)T \)
61 \( 1 + (0.899 - 0.437i)T \)
67 \( 1 + (-0.885 + 0.464i)T \)
71 \( 1 + (0.491 + 0.871i)T \)
73 \( 1 + (0.977 + 0.209i)T \)
83 \( 1 + (-0.180 + 0.983i)T \)
89 \( 1 + (-0.992 - 0.120i)T \)
97 \( 1 + (-0.326 + 0.945i)T \)
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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

−18.5920508727826349980375632994, −18.13261472647013788183100437768, −17.07860867395144249096312325942, −16.47956061906880983532086072386, −16.25746515558719738537429652898, −15.33359897285769773550028901275, −14.93390088147465033488631078568, −14.13738208537716793193883532844, −13.192991879168980818953195509995, −12.710591057048221025984862957973, −12.026819128178356673372772965688, −11.17304333077459529129224949385, −10.2101602726621356387174495740, −9.33670476945593940371791592639, −9.053133674846236067088564303327, −8.302524320123737669254976697674, −7.550376642176865065264636046278, −6.77273720046283748186715192484, −6.19136249207240837238844377634, −5.439771975817785141158357638705, −4.46607172984264474492125017802, −4.033130121520195685676579427027, −3.10642171564653756467512765694, −1.66673727698239701939470151217, −0.91962004176407854213265186088, 0.473574674629975977613665017600, 1.073877490577022629008586932772, 2.485606121942955752630393781483, 3.127458834485315514434679163520, 3.69546520310680684302633561793, 4.24775183843413199886413144090, 5.37162994910261525613036526428, 6.457151061977670162082270543910, 6.965840370331557923881698912864, 7.9615638462707315290821511051, 8.4894703426919323274522402646, 9.37734421035095217504816830643, 9.96993945515774776968610623479, 10.817242439193923537952566006341, 11.27615032205388774722821955449, 11.74550623798285189850320561093, 12.7832154167090129806151582175, 13.33839062444047794298303481865, 13.83296254060970431831786073864, 14.79351151664313328408886164304, 15.54244403650314765970747841116, 16.268161244318880091953368292723, 16.97831786032810594481244013996, 17.63870549770548644141999068845, 18.47916417750829683764098809361

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