Properties

Label 1-4029-4029.2348-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.928 + 0.370i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.935 + 0.354i)2-s + (0.748 − 0.663i)4-s + (0.787 + 0.616i)5-s + (0.954 + 0.297i)7-s + (−0.464 + 0.885i)8-s + (−0.954 − 0.297i)10-s + (−0.787 + 0.616i)11-s + (0.748 − 0.663i)13-s + (−0.998 + 0.0603i)14-s + (0.120 − 0.992i)16-s + (0.239 − 0.970i)19-s + (0.998 − 0.0603i)20-s + (0.517 − 0.855i)22-s + (−0.707 − 0.707i)23-s + (0.239 + 0.970i)25-s + (−0.464 + 0.885i)26-s + ⋯
L(s)  = 1  + (−0.935 + 0.354i)2-s + (0.748 − 0.663i)4-s + (0.787 + 0.616i)5-s + (0.954 + 0.297i)7-s + (−0.464 + 0.885i)8-s + (−0.954 − 0.297i)10-s + (−0.787 + 0.616i)11-s + (0.748 − 0.663i)13-s + (−0.998 + 0.0603i)14-s + (0.120 − 0.992i)16-s + (0.239 − 0.970i)19-s + (0.998 − 0.0603i)20-s + (0.517 − 0.855i)22-s + (−0.707 − 0.707i)23-s + (0.239 + 0.970i)25-s + (−0.464 + 0.885i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.928 + 0.370i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (2348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.928 + 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.426350728 + 0.2738725509i\)
\(L(\frac12)\) \(\approx\) \(1.426350728 + 0.2738725509i\)
\(L(1)\) \(\approx\) \(0.8971108430 + 0.1809605765i\)
\(L(1)\) \(\approx\) \(0.8971108430 + 0.1809605765i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.935 + 0.354i)T \)
5 \( 1 + (0.787 + 0.616i)T \)
7 \( 1 + (0.954 + 0.297i)T \)
11 \( 1 + (-0.787 + 0.616i)T \)
13 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (0.239 - 0.970i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.983 - 0.180i)T \)
31 \( 1 + (-0.911 - 0.410i)T \)
37 \( 1 + (0.855 - 0.517i)T \)
41 \( 1 + (0.616 - 0.787i)T \)
43 \( 1 + (-0.992 + 0.120i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (-0.464 + 0.885i)T \)
59 \( 1 + (-0.663 + 0.748i)T \)
61 \( 1 + (-0.517 - 0.855i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (0.954 - 0.297i)T \)
73 \( 1 + (0.998 - 0.0603i)T \)
83 \( 1 + (0.663 + 0.748i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (0.855 - 0.517i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.363516907681466675236109707454, −17.911176393617219319185347074310, −17.10329307159813415930881530963, −16.47079827701365378949282835249, −16.15493527085354462252249475477, −15.1407876872430011601081926380, −14.22261432213824584636318526958, −13.57469937750535337350011872546, −12.94603462335813198240033673319, −12.10511334296999664666671061205, −11.347193568589844037711501732, −10.84071480274078666958538018211, −10.10950341011958044952746132504, −9.38102908026158620102165244551, −8.77293400427563564830151047389, −7.97630436408546127855976759208, −7.67290814203773855843206411015, −6.455769828495959960153909762806, −5.81121680782896041385054901650, −5.03538146548704239882877372739, −4.000183575039954668328329803, −3.257803675014744230098579020162, −2.019471459753012748226764291362, −1.68018468334105099632904969492, −0.82585190184088911362327597481, 0.69179015327368424993944021265, 1.880911080965962129558640349478, 2.23525280636799526994269411860, 3.10749637049050690853615662340, 4.44658309561966693485937553187, 5.46725180270163653649879545482, 5.75044020548646704599759307929, 6.67693036302720063270518091704, 7.54590807001224821646977584944, 7.888612975868272054899778460624, 8.892413844545760869030053133238, 9.39742667362620669916562225474, 10.26309035563412954268649798022, 10.875760449263849245905926875189, 11.1859931786785668638551728364, 12.2534624673168437287961192638, 13.139811754293476590118317915257, 13.91870287998600654364927209288, 14.62572394699930661532032440110, 15.25582799695146173302740278907, 15.64996235832541369468982468958, 16.68036412097673220503193917340, 17.32949982013193407752659108293, 18.02803102229255011751809225946, 18.28463969793462126373111192049

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