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Properties

Label 1-4033-4033.608-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.852 + 0.522i$
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 + (0.173 + 0.984i)3-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + 10-s + 11-s + (−0.939 + 0.342i)12-s + (−0.939 − 0.342i)13-s + 14-s + (0.766 + 0.642i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)3-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + 10-s + 11-s + (−0.939 + 0.342i)12-s + (−0.939 − 0.342i)13-s + 14-s + (0.766 + 0.642i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (608, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.402335449 + 0.9588047980i\)
\(L(\frac12)\) \(\approx\) \(3.402335449 + 0.9588047980i\)
\(L(1)\) \(\approx\) \(1.829626750 + 0.7948557763i\)
\(L(1)\) \(\approx\) \(1.829626750 + 0.7948557763i\)

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 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + T \)
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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

−18.66672782257732068705276420670, −17.75521708326625421329509503995, −17.45860759958439761909724946899, −16.44162623189957366663716692437, −15.16069289415075043414328948813, −14.70314277107026206717391839709, −14.12515181976947361434182122737, −13.89155818536822972987224343083, −12.79734946991928688508573310725, −12.254621636288610249810520800912, −11.76098628335892586787839183470, −11.06244107846764059354662430537, −10.294477692793631190779109308965, −9.346848487595497949388542268626, −8.94240741662340174242701210543, −7.71581730875769536307105205833, −7.05805162642109202758031942730, −6.30935489340056602464909800232, −5.63061791005048296026820027433, −5.18788873609065447181263778324, −3.88054012047971172808085198586, −3.23474988371898478871852691363, −2.28640940054330501341453233625, −1.74753476382069110507300272247, −1.30559886917241045112344816753, 0.68390498373534022475211265122, 2.09517896931992600156836600687, 2.77296626014544968600050544809, 3.813345391691887135504480880775, 4.52517061012589596505001920548, 4.89427430959406106368772754544, 5.61205107064187790614707957545, 6.36685882671734983075038015468, 7.397529187528788582009694209916, 7.95908863588265993390796038104, 8.856389749251714189780651995419, 9.48478498218813397193641254020, 10.02691828194059356859985331995, 11.19857048181557848430595210701, 11.61882478879233231728380226084, 12.4472713831741224374581853529, 13.37712210691376746216453217637, 13.949709640978776872883838972860, 14.46551706055559531652690341726, 14.92150384121935141019607634587, 15.84990727894413783893562738142, 16.623515093054339704082073120624, 16.82040887735750766068199732675, 17.60302890443994925085642819979, 18.02688026845725043905542360962

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