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Properties

Label 1-4033-4033.1052-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.780 - 0.625i$
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  + (−0.173 + 0.984i)2-s + (−0.939 + 0.342i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)5-s + (−0.173 − 0.984i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + 12-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.766 − 0.642i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 + 0.342i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)5-s + (−0.173 − 0.984i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + 12-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.766 − 0.642i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.780 - 0.625i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1052, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.780 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1247300717 - 0.04379276959i\)
\(L(\frac12)\) \(\approx\) \(0.1247300717 - 0.04379276959i\)
\(L(1)\) \(\approx\) \(0.3714342519 + 0.2671668793i\)
\(L(1)\) \(\approx\) \(0.3714342519 + 0.2671668793i\)

Euler product

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

−18.76795600765533518841144247773, −18.05968261854057867650804356296, −17.11539472320785331346370992603, −16.49621685592637865621380715095, −16.28504853819860848704094409088, −15.311226452952426254281922984, −14.03327398781401227046753566556, −13.63325877855601959869451845104, −12.723806100505740758865236471448, −12.22028155336465736212616097000, −11.77947250542451693865645925389, −11.11502368299088927116845790089, −10.36375081607654210150422054274, −9.762311341358073527823389659624, −9.008025779997681415615548609142, −7.89387809560820960869945070372, −7.64149093085188761783118395282, −6.55496143584844200697594801651, −5.77032721722357690693156029408, −4.84607101352002903090415138980, −4.2882378896784887225007086697, −3.38136255888400431438949809766, −2.805130687844443248478789148865, −1.4559012457699417505572013093, −0.75622137515562364250103925860, 0.08133632912532753069851148884, 1.03392608116223422160100465066, 2.67454249781157415853662024218, 3.64791628020721561875236801523, 4.256282088985604344106015079523, 5.190971358319394090988754622492, 5.605477008507384999175377017, 6.54567766958444100089593936094, 7.17162753618148147924273874872, 7.61886116470937869297545264301, 8.52793401241236058496513644096, 9.57827363212791275871085023188, 10.164699652645558332194213740742, 10.33907993638656370007873556456, 11.92935558232103125489036152721, 12.08361159217172179905710210349, 12.72556051193572884926960887722, 13.79271094163405062180475545872, 14.59156920001776663895016656954, 15.441693741549344296891782918097, 15.62829144552839946472393005271, 16.217396606574732099209229867917, 16.95452499307859127963475064346, 17.58854713754845496758160724051, 18.35616463183761536259834534147

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