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Properties

Label 1-4029-4029.2150-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.810 + 0.586i$
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  + (−0.464 + 0.885i)2-s + (−0.568 − 0.822i)4-s + (−0.410 + 0.911i)5-s + (0.616 + 0.787i)7-s + (0.992 − 0.120i)8-s + (−0.616 − 0.787i)10-s + (0.410 + 0.911i)11-s + (−0.568 − 0.822i)13-s + (−0.983 + 0.180i)14-s + (−0.354 + 0.935i)16-s + (−0.663 + 0.748i)19-s + (0.983 − 0.180i)20-s + (−0.998 − 0.0603i)22-s + (0.707 − 0.707i)23-s + (−0.663 − 0.748i)25-s + (0.992 − 0.120i)26-s + ⋯
L(s)  = 1  + (−0.464 + 0.885i)2-s + (−0.568 − 0.822i)4-s + (−0.410 + 0.911i)5-s + (0.616 + 0.787i)7-s + (0.992 − 0.120i)8-s + (−0.616 − 0.787i)10-s + (0.410 + 0.911i)11-s + (−0.568 − 0.822i)13-s + (−0.983 + 0.180i)14-s + (−0.354 + 0.935i)16-s + (−0.663 + 0.748i)19-s + (0.983 − 0.180i)20-s + (−0.998 − 0.0603i)22-s + (0.707 − 0.707i)23-s + (−0.663 − 0.748i)25-s + (0.992 − 0.120i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.810 + 0.586i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (2150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.810 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8933136679 + 0.2893062145i\)
\(L(\frac12)\) \(\approx\) \(0.8933136679 + 0.2893062145i\)
\(L(1)\) \(\approx\) \(0.6616100726 + 0.3741918285i\)
\(L(1)\) \(\approx\) \(0.6616100726 + 0.3741918285i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.464 + 0.885i)T \)
5 \( 1 + (-0.410 + 0.911i)T \)
7 \( 1 + (0.616 + 0.787i)T \)
11 \( 1 + (0.410 + 0.911i)T \)
13 \( 1 + (-0.568 - 0.822i)T \)
19 \( 1 + (-0.663 + 0.748i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.855 - 0.517i)T \)
31 \( 1 + (-0.297 - 0.954i)T \)
37 \( 1 + (-0.0603 - 0.998i)T \)
41 \( 1 + (-0.911 - 0.410i)T \)
43 \( 1 + (-0.935 + 0.354i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (0.992 - 0.120i)T \)
59 \( 1 + (0.822 + 0.568i)T \)
61 \( 1 + (0.998 - 0.0603i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (0.616 - 0.787i)T \)
73 \( 1 + (0.983 - 0.180i)T \)
83 \( 1 + (-0.822 + 0.568i)T \)
89 \( 1 + (0.120 - 0.992i)T \)
97 \( 1 + (-0.0603 - 0.998i)T \)
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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

−18.64128440156670189092726265949, −17.58091898934238237492497503693, −16.98921709081534317432369330791, −16.77283943616594238508657635844, −15.95605645208862020672298265759, −14.92003602619725520172468985199, −14.089087511609866485235280923351, −13.43725443807332750695186399015, −12.92107641271074519941975504749, −12.02830467758470435385890736727, −11.39554356181600990139972750108, −11.087311227508997261654713014244, −10.11941789180032968509348321673, −9.34621678442494307182414900684, −8.66316277588171463159769111600, −8.29776770996674026951041737718, −7.29789127334257761820905233851, −6.78929201980759217529963222527, −5.20906002534148027179119204404, −4.85252715584113842127354995819, −3.90429201868243628213426339854, −3.50467205698607557303154424001, −2.27745073677594317097281853179, −1.39492850970977888600043272236, −0.8323910015075727557523418400, 0.40875020337125546094220991298, 1.89854974984146643539856428966, 2.35569694651644527848005474283, 3.67279227504261489926768386536, 4.40672803402700449590735074801, 5.306167847844253477165889383016, 5.87878432990295317787489171631, 6.80401432912205536244046194286, 7.30597865385522979969007370343, 8.04470222148607740473401252543, 8.58623388646685057338543590119, 9.528255317858509530488627155323, 10.13801524153469154056219555164, 10.81500101867454112991913629219, 11.579517855308839788490364708221, 12.388568405435685899484212448699, 13.10609516364360037883581202199, 14.22285210906057346035868053626, 14.731343377292719127973281366328, 15.17993622112306164756564147073, 15.46613325320870973934554318708, 16.69777092404371777782450702283, 17.1262240705105094581913872647, 17.9529912801751822581959962418, 18.40497691782203811907524583128

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