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Properties

Label 1-4033-4033.105-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.518 + 0.855i$
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  + (0.866 − 0.5i)2-s + (−0.766 + 0.642i)3-s + (0.5 − 0.866i)4-s + (0.984 − 0.173i)5-s + (−0.342 + 0.939i)6-s + (0.173 + 0.984i)7-s − i·8-s + (0.173 − 0.984i)9-s + (0.766 − 0.642i)10-s + (−0.766 − 0.642i)11-s + (0.173 + 0.984i)12-s + (0.984 − 0.173i)13-s + (0.642 + 0.766i)14-s + (−0.642 + 0.766i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.766 + 0.642i)3-s + (0.5 − 0.866i)4-s + (0.984 − 0.173i)5-s + (−0.342 + 0.939i)6-s + (0.173 + 0.984i)7-s − i·8-s + (0.173 − 0.984i)9-s + (0.766 − 0.642i)10-s + (−0.766 − 0.642i)11-s + (0.173 + 0.984i)12-s + (0.984 − 0.173i)13-s + (0.642 + 0.766i)14-s + (−0.642 + 0.766i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.518 + 0.855i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6788724881 + 1.205435899i\)
\(L(\frac12)\) \(\approx\) \(0.6788724881 + 1.205435899i\)
\(L(1)\) \(\approx\) \(1.436190103 - 0.09985505219i\)
\(L(1)\) \(\approx\) \(1.436190103 - 0.09985505219i\)

Euler product

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

−17.81911797084351110242000648636, −17.369144766012232598993706550663, −16.792235784717509327327898444962, −16.11666723155199042205806343728, −15.32091156823638983838018636324, −14.57448232611826978043842912365, −13.506377306747214414894209329198, −13.421791454263983369862445206714, −13.04566944556280461717465703410, −11.96447766522152082001170431258, −11.282455485283756019813372652823, −10.60431309136403229377419017055, −10.132101863684626611786410068511, −8.76172027759315848950665464937, −8.0973333689928344329987731840, −7.1623168316608246144535046826, −6.57420725755832743525660319423, −6.37634063605106985374854707186, −5.16969724314043676082648636858, −4.85049309200436802021506590103, −4.04261287976116955584225057235, −2.77100441045289774616968406530, −2.17230380601738381514877118680, −1.28642133173118653989799056590, −0.147180278503194515448098298758, 1.11882561804733747150348015814, 1.79416000760734577512800658595, 2.83065875634882410412593359343, 3.387900843349997520307090123972, 4.54904402167084551744740141209, 5.037978950909486986077879572633, 5.748113536013132615757134106623, 6.19346678098091437706445724497, 6.73973925955678850816327423420, 8.46904493972815495221196720349, 8.83998409797373243906997300471, 9.8719435341098917918277955977, 10.404326236588607764210232081234, 11.035177481127336122170120061, 11.54552407610336958261491254200, 12.46822068221483237302519001391, 13.03795420863801232815994476831, 13.509811037388160285966470534058, 14.53843446783638145073231283560, 15.07730261162831918933808212176, 15.83120585736471792742497120219, 16.18485468395114312591238632156, 17.222962964704697627066218015594, 17.846852830324908366864040651306, 18.56826693831561042456965980584

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