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Properties

Label 1-4033-4033.1457-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.838 - 0.544i$
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 + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (0.802 − 0.597i)5-s + (0.973 + 0.230i)6-s + (0.396 − 0.918i)7-s + (0.5 − 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (−0.396 + 0.918i)12-s + (−0.396 + 0.918i)13-s + (0.835 + 0.549i)14-s + (−0.549 − 0.835i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (0.802 − 0.597i)5-s + (0.973 + 0.230i)6-s + (0.396 − 0.918i)7-s + (0.5 − 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (−0.396 + 0.918i)12-s + (−0.396 + 0.918i)13-s + (0.835 + 0.549i)14-s + (−0.549 − 0.835i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.838 - 0.544i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.838 - 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.496132423 - 0.4434757642i\)
\(L(\frac12)\) \(\approx\) \(1.496132423 - 0.4434757642i\)
\(L(1)\) \(\approx\) \(1.029423873 + 0.02666609498i\)
\(L(1)\) \(\approx\) \(1.029423873 + 0.02666609498i\)

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 \( 1 + (0.0581 - 0.998i)T \)
5 \( 1 + (0.802 - 0.597i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (-0.448 + 0.893i)T \)
13 \( 1 + (-0.396 + 0.918i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.998 + 0.0581i)T \)
31 \( 1 + (-0.116 + 0.993i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.727 - 0.686i)T \)
53 \( 1 + (-0.116 - 0.993i)T \)
59 \( 1 + (0.835 - 0.549i)T \)
61 \( 1 + (-0.957 + 0.286i)T \)
67 \( 1 + (0.802 + 0.597i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.893 - 0.448i)T \)
79 \( 1 + (-0.396 - 0.918i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (0.957 - 0.286i)T \)
97 \( 1 + (0.802 - 0.597i)T \)
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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

−18.59082060037380399948248865235, −18.062771388095542863108339027936, −17.16013451102896969955630639746, −16.76161532038699812049018281212, −15.75871860516907372533953739121, −14.948565580493827630564609524973, −14.40905086909790434190949432713, −13.86807364631594714527684352210, −12.9705915456330575236026379719, −12.232233647688238671498254957607, −11.46558194889146616646081495673, −10.806750440360285196920851202334, −10.27132250705705679840776509481, −9.78621432629915458272232140479, −8.97109542027867744952352400425, −8.36887442705492092716109097038, −7.750477479618021043128398573198, −6.196026905808489016258582466851, −5.58136024966150311792022538363, −5.06792655637604141833243967044, −4.182995402833270412865736066301, −3.045544260733201239522231070125, −2.827578359677379475635225511777, −2.11984416211384197627612361145, −0.7987874746161344997622969130, 0.60978080203855128308822641727, 1.56250627669556950283831911547, 2.00732051506903143914194624109, 3.42502397352177767824238445967, 4.57822988071511465808592505382, 4.97858835273327528055730351002, 5.85228313610304483818641675318, 6.65864325258706555529113449793, 7.1395897645536675108479034286, 7.786368015420796515829375422547, 8.558311247620951603866510042399, 9.1238748448933044154608014181, 10.05155795970998328733816946162, 10.52562731886707669989802520028, 11.76954951747779599334096722034, 12.52996501479236589157438235985, 13.18521987765161598404694544840, 13.60583078498608124236296953779, 14.40632418572820660115691667395, 14.691198316741618874153083689316, 15.82570796236571201147417156013, 16.61906142810067288240513452925, 17.30357657234391834657431513154, 17.472227425312105340186669059190, 18.08016698043094018496315574920

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