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Properties

Label 1-4033-4033.184-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.984 + 0.176i$
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  + (0.5 + 0.866i)2-s + (0.173 + 0.984i)3-s + (−0.5 + 0.866i)4-s + (0.939 + 0.342i)5-s + (−0.766 + 0.642i)6-s + (−0.939 − 0.342i)7-s − 8-s + (−0.939 + 0.342i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (−0.939 − 0.342i)12-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.173 + 0.984i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.173 + 0.984i)3-s + (−0.5 + 0.866i)4-s + (0.939 + 0.342i)5-s + (−0.766 + 0.642i)6-s + (−0.939 − 0.342i)7-s − 8-s + (−0.939 + 0.342i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (−0.939 − 0.342i)12-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.173 + 0.984i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.984 + 0.176i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.984 + 0.176i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2286285973 + 2.569612594i\)
\(L(\frac12)\) \(\approx\) \(0.2286285973 + 2.569612594i\)
\(L(1)\) \(\approx\) \(0.9416071627 + 1.202726240i\)
\(L(1)\) \(\approx\) \(0.9416071627 + 1.202726240i\)

Euler product

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

−18.30399268899243848024353196733, −17.71524559783017334459605693378, −17.11814159099512662271958506329, −15.94109089227337287383081767044, −15.34302426670013240157842321763, −14.360589256275324883613244948230, −13.72290688948359520191737422682, −13.36626031163174390444407603049, −12.594642570431891196420081254517, −12.31950905162073561532154915569, −11.46084774925778261011933571885, −10.62713280019116783306909742785, −9.75725998885410262427289717578, −9.18066449051386898201025624880, −8.82179003136887052275726962944, −7.49159491618627699509950308003, −6.73524464084342458994676569336, −6.0597307177210402767936099018, −5.4264437074869440396946058246, −4.747595894206913469998528641575, −3.41694256271563009864219208860, −2.939296168517842405156482762994, −2.177538666874202418338694400545, −1.37533297938125313547165669721, −0.69493044072941337490152227106, 1.027075207626271989252734188544, 2.63047139902798429844315498091, 3.17109290776161655145021557695, 3.88297211297059708634555432504, 4.478081504606556212312664221895, 5.73605762653669607893364088198, 5.99694079453920774873397870802, 6.43411297657236125954694206021, 7.64173202621448483766103887613, 8.39051266047548055864915725545, 9.12824290978569685527691064946, 9.633656186320625843022692593, 10.44768910022409372458182219643, 11.03729395663601068860264224395, 12.05866827601677318155874343793, 12.97337284801467321314906687554, 13.646775360150187302159699730258, 14.01895935492148900928137848783, 14.687332423483585319064370401446, 15.39086049690997436132123236856, 16.30724498400331469591472752206, 16.44690537333259754255653404246, 17.108520924760729601439738377496, 17.83896788915238408716330133790, 18.8850044530860742686181203926

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