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Properties

Label 1-4033-4033.432-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.548 + 0.836i$
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  + (0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + 12-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)14-s + (−0.173 − 0.984i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + 12-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)14-s + (−0.173 − 0.984i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.548 + 0.836i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.548 + 0.836i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8826991923 + 1.633590092i\)
\(L(\frac12)\) \(\approx\) \(0.8826991923 + 1.633590092i\)
\(L(1)\) \(\approx\) \(1.708627356 + 0.2057396761i\)
\(L(1)\) \(\approx\) \(1.708627356 + 0.2057396761i\)

Euler product

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

−18.089329424235787944463731829217, −17.72243356508108346005031412366, −16.810448141536606318333356790131, −15.84265421984880700889156622121, −15.16187198125727879146439031118, −14.941139864013243922649495665073, −14.17248614399847854463559898790, −13.39673577253323979909362957334, −13.0781183807496629542960903268, −12.21142899695636400976501656685, −11.531345163214914195838647352, −10.67424803943690799477902798730, −10.38531697637990073046711118150, −8.67492226543823349595255647343, −8.24626441561571513001707237365, −7.57594894432450814151116064740, −7.19604004695538727049328969963, −6.35777213302450466980106337672, −5.55090380603066629831529407988, −4.49839303174808785347083478743, −3.93853275681447577158316234308, −3.16942412433855515302160045573, −2.483047680219960818342864580553, −1.764436235841491944625126021344, −0.272366503368497746188901269, 1.58977380480552780645019228073, 2.16759520416311623751084501055, 2.86837933847654067662936933300, 3.93326807145554095039111446247, 4.44357908489808073088295236804, 4.89996345887251042645255765413, 5.637680518659489165650924493688, 6.738396371873330927204190461356, 7.77937313731959757383972288101, 8.10589122275016845913022066440, 9.10415952534833772959260296813, 9.63898577488052087921175697360, 10.720412509409754005708612442617, 11.10705223940451049292093968457, 12.07027226820447328584286831619, 12.395142056372078485292095330204, 13.35696114638258379505785651983, 13.99055858752498433707905271250, 14.58468379239410104828144717959, 15.39681988575208890005909138483, 15.65050122179980909988037396325, 16.2715190698639453310189464212, 17.102328868420241681224842893004, 18.33243623993877169775746554427, 18.936933435393652107281127456532

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