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Properties

Label 1-4033-4033.1031-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.266 + 0.963i$
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 + (0.0581 + 0.998i)3-s + (0.766 − 0.642i)4-s + (−0.116 + 0.993i)5-s + (−0.396 − 0.918i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.230 − 0.973i)10-s + (0.230 − 0.973i)11-s + (0.686 + 0.727i)12-s + (0.396 + 0.918i)13-s + (−0.286 + 0.957i)14-s + (−0.998 − 0.0581i)15-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.0581 + 0.998i)3-s + (0.766 − 0.642i)4-s + (−0.116 + 0.993i)5-s + (−0.396 − 0.918i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.230 − 0.973i)10-s + (0.230 − 0.973i)11-s + (0.686 + 0.727i)12-s + (0.396 + 0.918i)13-s + (−0.286 + 0.957i)14-s + (−0.998 − 0.0581i)15-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.266 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024768400 + 0.7800957925i\)
\(L(\frac12)\) \(\approx\) \(1.024768400 + 0.7800957925i\)
\(L(1)\) \(\approx\) \(0.7386099844 + 0.3769499336i\)
\(L(1)\) \(\approx\) \(0.7386099844 + 0.3769499336i\)

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 \( 1 + (0.0581 + 0.998i)T \)
5 \( 1 + (-0.116 + 0.993i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (0.230 - 0.973i)T \)
13 \( 1 + (0.396 + 0.918i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (-0.549 + 0.835i)T \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.116 + 0.993i)T \)
53 \( 1 + (-0.727 + 0.686i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (-0.727 + 0.686i)T \)
67 \( 1 + (0.230 - 0.973i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.686 - 0.727i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (-0.396 - 0.918i)T \)
89 \( 1 + (0.998 - 0.0581i)T \)
97 \( 1 + (0.448 - 0.893i)T \)
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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

−18.36708891015652272036419690040, −17.70658060942412606570062210049, −17.22379057544092148119775626772, −16.5997975512539150827044345646, −15.65919262000907541653347823329, −15.09259993868864804607429638709, −14.26746646802286933534847668142, −13.122129652738767742377487630518, −12.646346580612820338251723864971, −12.21482792089685616550413563357, −11.53723084382521122883044575199, −10.93132804783607708315792656498, −9.74369186075277457246515962250, −9.25154467203904207740196274933, −8.49300630253402426921468235376, −7.94085618888261789057917007833, −7.47722816379964390894970580630, −6.59432458775581804123459485342, −5.49182134454433441282483984911, −5.21572105019804390712335017775, −3.73171943933986164807552594083, −2.98099556611112588168063392489, −1.96412444024606099764073024656, −1.45151502284552036401441097403, −0.76327616772769073405365367150, 0.7231030989276945783086551997, 1.699021310202931229900755293965, 2.93929973388250849737912547026, 3.37144967292247801476300617606, 4.34727503367772573065991374065, 5.29033328673607178495607467035, 6.028377433636204479371224241, 6.78803579306869695600227679547, 7.57472847479732917971315861698, 8.11631261574595720111255289355, 9.086238578835437951834814592079, 9.502456797538384453353932483241, 10.59240636569244690255324287613, 10.63015581728517603199522300025, 11.55280549890014317794854293220, 11.778326369352464377737411697952, 13.70891692489050331379540985478, 14.088684356202576413201673401328, 14.58030680641015146979011789592, 15.321390363876660362200684089863, 16.05170795905482566552068969303, 16.611411444205210957415910635152, 17.07717429622936734527897349957, 17.88761810798051350980407445609, 18.67368717825994781795538486010

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