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Properties

Label 1-4033-4033.1042-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.236 - 0.971i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.984 + 0.173i)2-s + (−0.396 + 0.918i)3-s + (0.939 + 0.342i)4-s + (−0.230 + 0.973i)5-s + (−0.549 + 0.835i)6-s + (−0.286 − 0.957i)7-s + (0.866 + 0.5i)8-s + (−0.686 − 0.727i)9-s + (−0.396 + 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.686 + 0.727i)12-s + (−0.957 + 0.286i)13-s + (−0.116 − 0.993i)14-s + (−0.802 − 0.597i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (−0.396 + 0.918i)3-s + (0.939 + 0.342i)4-s + (−0.230 + 0.973i)5-s + (−0.549 + 0.835i)6-s + (−0.286 − 0.957i)7-s + (0.866 + 0.5i)8-s + (−0.686 − 0.727i)9-s + (−0.396 + 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.686 + 0.727i)12-s + (−0.957 + 0.286i)13-s + (−0.116 − 0.993i)14-s + (−0.802 − 0.597i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.236 - 0.971i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.236 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2032091368 + 0.1596100056i\)
\(L(\frac12)\) \(\approx\) \(-0.2032091368 + 0.1596100056i\)
\(L(1)\) \(\approx\) \(1.164639695 + 0.7787467115i\)
\(L(1)\) \(\approx\) \(1.164639695 + 0.7787467115i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (-0.396 + 0.918i)T \)
5 \( 1 + (-0.230 + 0.973i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (0.396 + 0.918i)T \)
13 \( 1 + (-0.957 + 0.286i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.802 - 0.597i)T \)
31 \( 1 + (0.957 + 0.286i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.893 - 0.448i)T \)
53 \( 1 + (-0.973 - 0.230i)T \)
59 \( 1 + (-0.802 - 0.597i)T \)
61 \( 1 + (-0.448 - 0.893i)T \)
67 \( 1 + (-0.286 + 0.957i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.396 - 0.918i)T \)
79 \( 1 + (-0.727 + 0.686i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (-0.549 + 0.835i)T \)
97 \( 1 + (-0.727 - 0.686i)T \)
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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

−17.680368172985499254739067953199, −16.91642099111030727157710351621, −16.39489208461160595353802400927, −15.74469431822245236963992315020, −14.93809416834557463393800278930, −14.188136327925648712730385319221, −13.51830293448508149162286476033, −12.755849979905213892317058858146, −12.38425890573825094174552971725, −11.89319864206001884276056115171, −11.287877230267225477729187531118, −10.37660623982266332870774671374, −9.415393604321640685854029496410, −8.475461081835121807103727900806, −7.97318980227814883061099987270, −7.04025149819389424766626663181, −6.13049101012156897498127266396, −5.85638071407892167879982832937, −4.95375867232123532724518991980, −4.48595999961662741107849171960, −3.10849634218704605796168218080, −2.71892806870860520409917553290, −1.64651943517352837616375556354, −0.95522702530190047233031882693, −0.02761627128580360558263680465, 1.41519575084743374973547997410, 2.60311479087979555604060221250, 3.28390374232446716733757240491, 3.94077382160937308041724687275, 4.51816035946041578273515113695, 5.24896159679279846713756677595, 6.15957897831388010123354078108, 6.81103746866863654493094999919, 7.392104280140440719106422893789, 8.04630530382804995894134594615, 9.73163512920314907245351452066, 9.86218113640541393890043759023, 10.58234865286832418980253442952, 11.44226075601871155628224044476, 11.94123486821058776634730694285, 12.46632983577833213842620615568, 13.72505299981240323328851619017, 14.30300672553586991665972594640, 14.518955119643493051864093937, 15.57676727528587638727754543168, 15.79349772031864986480474409119, 16.72203219901082022387405426697, 17.26998881826838332554337243727, 17.810933953863563062518637117276, 19.07779926449495152551032187506

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