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Properties

Label 1-4033-4033.1059-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.474 + 0.880i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.342 + 0.939i)2-s + (−0.893 − 0.448i)3-s + (−0.766 − 0.642i)4-s + (−0.918 + 0.396i)5-s + (0.727 − 0.686i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.0581 − 0.998i)10-s + (0.0581 − 0.998i)11-s + (0.396 + 0.918i)12-s + (0.116 − 0.993i)13-s + (0.448 − 0.893i)14-s + (0.998 + 0.0581i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (−0.893 − 0.448i)3-s + (−0.766 − 0.642i)4-s + (−0.918 + 0.396i)5-s + (0.727 − 0.686i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.0581 − 0.998i)10-s + (0.0581 − 0.998i)11-s + (0.396 + 0.918i)12-s + (0.116 − 0.993i)13-s + (0.448 − 0.893i)14-s + (0.998 + 0.0581i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.474 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04031379125 + 0.06749326198i\)
\(L(\frac12)\) \(\approx\) \(0.04031379125 + 0.06749326198i\)
\(L(1)\) \(\approx\) \(0.4242764427 + 0.05217391340i\)
\(L(1)\) \(\approx\) \(0.4242764427 + 0.05217391340i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 + (-0.893 - 0.448i)T \)
5 \( 1 + (-0.918 + 0.396i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (0.0581 - 0.998i)T \)
13 \( 1 + (0.116 - 0.993i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (0.549 + 0.835i)T \)
31 \( 1 + (-0.918 - 0.396i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (-0.286 - 0.957i)T \)
53 \( 1 + (-0.993 - 0.116i)T \)
59 \( 1 + (-0.998 - 0.0581i)T \)
61 \( 1 + (-0.727 + 0.686i)T \)
67 \( 1 + (-0.597 - 0.802i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.0581 - 0.998i)T \)
79 \( 1 + (0.918 + 0.396i)T \)
83 \( 1 + (-0.0581 - 0.998i)T \)
89 \( 1 + (0.957 + 0.286i)T \)
97 \( 1 + (0.116 - 0.993i)T \)
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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

−18.130026237896655665889166250972, −17.37443541857848330186287825932, −16.79063766567314166434570480047, −16.12243225866300327386626633984, −15.676058750295219836744739278241, −14.79429609098919416745370899587, −13.759858987290671269370031863979, −12.82674221151936615864470886968, −12.35864479761142316118113826686, −11.98989486282681812368622029334, −11.11742878229424234807194873767, −10.69039696971811716027079347719, −9.706227462703393777702264483728, −9.29493122829159819340498334166, −8.69963750451436874139653841165, −7.47352759153354119823370597156, −6.991731867263808116438947387182, −6.04268278421101194988977356028, −4.94315195090410080754983438258, −4.3774788489717187120187001248, −3.86394478666188488878359431446, −3.04689296517326551551393359086, −1.92360282673491467371276476943, −0.99950222856000485516421319298, −0.04102583343965773685936610276, 0.43916965677317328688826184224, 1.19022422011764641931447367295, 2.80487391568317806515805769919, 3.52918254200944887050631327717, 4.565624240564791334700465515238, 5.204335069937799660706972927921, 6.2433183685677538686962697851, 6.42352524412152739364283346699, 7.348766012145315150166145314116, 7.72212516964113776382402117846, 8.72409331422796262132445736885, 9.30696170776902343241632392478, 10.502730300786484613035876245245, 10.80798921048470656358329281762, 11.47531808600497622705631631793, 12.560469068533984388574078055561, 13.15038038263454161944217162633, 13.57152217825198577629038434147, 14.67771887449331523569761751346, 15.42606576625425585457420844456, 15.92249440574550190048259886304, 16.405433276599915843856087763870, 17.05808192013666917479247966462, 17.92583243857205374798980598912, 18.303952421511368537167726678415

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