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Properties

Label 1-4033-4033.847-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.902 - 0.430i$
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  + (−0.173 + 0.984i)2-s + (−0.686 − 0.727i)3-s + (−0.939 − 0.342i)4-s + (−0.0581 − 0.998i)5-s + (0.835 − 0.549i)6-s + (0.893 + 0.448i)7-s + (0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.993 + 0.116i)10-s + (0.993 − 0.116i)11-s + (0.396 + 0.918i)12-s + (0.835 − 0.549i)13-s + (−0.597 + 0.802i)14-s + (−0.686 + 0.727i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.686 − 0.727i)3-s + (−0.939 − 0.342i)4-s + (−0.0581 − 0.998i)5-s + (0.835 − 0.549i)6-s + (0.893 + 0.448i)7-s + (0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.993 + 0.116i)10-s + (0.993 − 0.116i)11-s + (0.396 + 0.918i)12-s + (0.835 − 0.549i)13-s + (−0.597 + 0.802i)14-s + (−0.686 + 0.727i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (847, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.902 - 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.164259397 - 0.2632500782i\)
\(L(\frac12)\) \(\approx\) \(1.164259397 - 0.2632500782i\)
\(L(1)\) \(\approx\) \(0.8320770513 + 0.05130874071i\)
\(L(1)\) \(\approx\) \(0.8320770513 + 0.05130874071i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.686 - 0.727i)T \)
5 \( 1 + (-0.0581 - 0.998i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.993 - 0.116i)T \)
13 \( 1 + (0.835 - 0.549i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.993 - 0.116i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.0581 - 0.998i)T \)
53 \( 1 + (-0.396 - 0.918i)T \)
59 \( 1 + (0.686 - 0.727i)T \)
61 \( 1 + (0.396 + 0.918i)T \)
67 \( 1 + (0.993 - 0.116i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (-0.597 - 0.802i)T \)
83 \( 1 + (-0.835 + 0.549i)T \)
89 \( 1 + (-0.686 - 0.727i)T \)
97 \( 1 + (0.973 - 0.230i)T \)
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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

−18.45658194248989787714596216842, −17.93780304997557629584454260934, −17.269169360135970284308098707363, −16.74221039030057542068764515167, −15.89081944404458049766012560897, −14.92342359335080863423071024231, −14.23940750530286372827143943245, −14.04302766103030130748971940010, −12.82345477218859096843555464319, −12.01464072598140406835182292235, −11.37692190865123144454858037828, −10.95211775966492920200368316675, −10.5838603003112695958195549463, −9.664489860458259023387647235636, −9.04769213799938436400405945854, −8.3343476700753623610134514321, −7.22490999604974711526167693600, −6.58561001780444165836372097473, −5.69096523972315924703195994797, −4.70155289583658316219246915605, −4.09396422923645926431599534657, −3.681639640437200147783337917416, −2.61905870837103198894439273177, −1.72507701989150422791944198947, −0.806795491141805102192117646246, 0.55502937601472496743287800095, 1.46910658677057302052048382103, 1.91775180505109710228088128358, 3.84917518528453845006661645790, 4.31300955226076216229173031282, 5.31626867334170172589099750707, 5.78175933515249636531124410826, 6.26860890099034630689061854788, 7.26139666451629855117264226734, 8.06432432919692085305743539931, 8.41098161624255251265477009937, 9.07496626050097989818750837390, 9.98014525313040931351726163741, 11.11545111805366901902889921621, 11.45613740588185846305843986900, 12.54072318496655200710764021454, 12.94596855801679263435505311909, 13.560199169181500216479875065, 14.441911119459865039673244974182, 15.10271337581439090031861825256, 15.80330504024234776854155902614, 16.60891225277443787626941715400, 17.11003459304538305312997429690, 17.58167614915417141228076621080, 18.09195901946946342810105586750

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