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Properties

Label 1-4033-4033.2312-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.893 - 0.448i$
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 + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (0.866 + 0.5i)5-s + (0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.984 − 0.173i)10-s + (−0.984 − 0.173i)11-s + (0.939 − 0.342i)12-s + (−0.5 − 0.866i)13-s + (0.173 + 0.984i)14-s + i·15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (0.866 + 0.5i)5-s + (0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.984 − 0.173i)10-s + (−0.984 − 0.173i)11-s + (0.939 − 0.342i)12-s + (−0.5 − 0.866i)13-s + (0.173 + 0.984i)14-s + i·15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.893 - 0.448i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1271945285 - 0.5369584333i\)
\(L(\frac12)\) \(\approx\) \(0.1271945285 - 0.5369584333i\)
\(L(1)\) \(\approx\) \(1.401753045 - 0.09638707829i\)
\(L(1)\) \(\approx\) \(1.401753045 - 0.09638707829i\)

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

−18.75890208435177282178480607169, −17.74680839275204005269802142994, −17.42276561731099658649187074998, −16.6195199954395116694113721255, −16.26362926779510979606636686388, −14.900334903982732430024621079033, −14.77657441761580060044876251435, −13.75752281522831658487833993133, −13.35685118741377243844266971838, −12.78339878208441988716120050307, −12.51162729659328584898793224590, −11.365236039627037676169199985475, −10.49456393437423050758998392299, −9.59375882385903706019101225169, −8.80278938801417772297105371012, −8.20046206654473504193015276163, −7.39541513062478948387340087793, −6.74613120292062974668322971744, −6.24282013857939023336229198608, −5.4782712262750733482514790489, −4.47115389921064233514116721523, −3.97490315534909171123605988482, −2.72442969257729487760620772249, −2.34224964471463136847576258030, −1.35449227839600983423116933293, 0.089449500214695164149665032174, 1.93364217507897728315373789523, 2.362770733289107033492935784493, 3.150826033261677262893901567751, 3.47267634280867301207861075631, 4.81045373245075512500595594400, 5.42312681168515172648216401611, 5.66490411676439913102551678313, 6.76618304618598377248266472412, 7.653902539177643992188578042360, 8.87346893329043433510508011437, 9.313786401086021139431956624006, 10.04197962801264478689350423947, 10.56869522114619446569508216630, 11.089907110194991293801442046259, 12.0589332614614401184506419714, 13.046047557765470176846731317, 13.21140584978686088442308329775, 14.12196374486639162905654290654, 14.85414304060339440760402008886, 15.16327538457085785712871192503, 15.90716341458567095466268542421, 16.547359366790735660491702609609, 17.71520677615243122849482894177, 18.316421435728610291066637243280

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