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Properties

Label 1-4033-4033.136-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.941 - 0.338i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.766 − 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s + (0.5 + 0.866i)5-s + (0.939 − 0.342i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.173 − 0.984i)10-s + (0.173 + 0.984i)11-s + (−0.939 − 0.342i)12-s + (0.5 − 0.866i)13-s + (−0.173 + 0.984i)14-s − 15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s + (0.5 + 0.866i)5-s + (0.939 − 0.342i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.173 − 0.984i)10-s + (0.173 + 0.984i)11-s + (−0.939 − 0.342i)12-s + (0.5 − 0.866i)13-s + (−0.173 + 0.984i)14-s − 15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.941 - 0.338i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.941 - 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9347995629 - 0.1627918282i\)
\(L(\frac12)\) \(\approx\) \(0.9347995629 - 0.1627918282i\)
\(L(1)\) \(\approx\) \(0.6935930726 + 0.02395795031i\)
\(L(1)\) \(\approx\) \(0.6935930726 + 0.02395795031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.38629336346806357015753057068, −17.95588219675302736768309290573, −17.07101672414026146661644582616, −16.5635258546978838678968665270, −15.9908510399792001377648782222, −15.55642537519822561196698197342, −14.08510931438944115425366552800, −13.86866019873396933956973786264, −13.193373996055073586736441780421, −12.21255854467707706373686641536, −11.61637197231072022928942449595, −11.07908207870471886527296281692, −9.98003835724227449967217261810, −9.17573535364122185154191343383, −8.83185720639492096403419658232, −8.175709654596890896638534142636, −7.177505016731795401612191791638, −6.56725229769223408232084659109, −5.953786885151348615099125054318, −5.29677891629655947821037364607, −4.831829159422270242046687397499, −3.17131086224800609223041521741, −2.2827491790212392661957925873, −1.338402237736036515431649071369, −0.84329810713549796694106923772, 0.50775116439694836790868759864, 1.527903301997266219301295091657, 2.6091698615301352978047043263, 3.35952664891000193664546534427, 3.90495577244547289607521981448, 4.71445158999675904074666591593, 5.89841203685527324949577921669, 6.52997126075210852157119610457, 7.239631829517787134296790866, 8.02876306732901387265171469661, 9.03586025223864761678705621690, 9.73467915808618402713927398206, 10.293387472055690539624736364859, 10.55267497182916931874917023056, 11.22820961083339794602238110082, 12.1234717407739184925064152324, 12.7802900120747091125059805293, 13.56478565946282222641032551224, 14.38942374804648734594843161162, 15.31011225302608443450615798592, 15.7058158137849804099459750843, 16.74223081504420763635553503354, 17.13823309043991689967252260222, 17.716474980557805398307340750300, 18.27950360429571301516796313728

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