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Properties

Label 1-4033-4033.3745-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.179 + 0.983i$
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 + (−0.973 + 0.230i)3-s + (0.173 − 0.984i)4-s + (−0.549 − 0.835i)5-s + (−0.597 + 0.802i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.957 − 0.286i)10-s + (0.957 − 0.286i)11-s + (0.0581 + 0.998i)12-s + (−0.0581 − 0.998i)13-s + (−0.686 − 0.727i)14-s + (0.727 + 0.686i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.973 + 0.230i)3-s + (0.173 − 0.984i)4-s + (−0.549 − 0.835i)5-s + (−0.597 + 0.802i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.957 − 0.286i)10-s + (0.957 − 0.286i)11-s + (0.0581 + 0.998i)12-s + (−0.0581 − 0.998i)13-s + (−0.686 − 0.727i)14-s + (0.727 + 0.686i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.179 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4483219748 - 0.3739105180i\)
\(L(\frac12)\) \(\approx\) \(-0.4483219748 - 0.3739105180i\)
\(L(1)\) \(\approx\) \(0.6535771605 - 0.6787804032i\)
\(L(1)\) \(\approx\) \(0.6535771605 - 0.6787804032i\)

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

−19.03191568950419686520556939059, −18.209917110709907969187181099881, −17.49514868486818179601250641752, −16.87286596710831694521423097896, −16.20385043025020177520817893453, −15.59203095412125128086221839504, −14.94409778661926184135392320786, −14.36050502952856930906045446090, −13.66006155092244197912725585921, −12.66306713530024986500294304381, −12.02718193982936406483353734618, −11.68650297040869100278489583519, −11.23445886856224308484680251453, −10.03242976526714565468852865727, −9.343400155176161785241847523915, −8.198271638288846168426195550535, −7.65471063925941099875335795990, −6.811067040372844657849995871093, −6.22253651742603822801674327297, −5.92742358622965915865409180229, −4.80694810968275289176206798293, −4.2094219459341385524177304775, −3.49295851399787159758954899118, −2.41849585641853741623137994167, −1.69378027072392504266347774987, 0.15561547370487937237975619824, 1.05975924976675196196510664592, 1.50300837171252634191262940497, 3.10848358200601383967122940325, 3.817447261059721095479606926330, 4.34728134240168128277344823974, 4.97859984787012324145527024014, 5.745801587652264737703706957812, 6.46419281431758516513227474774, 7.175256592305788349168922417640, 8.18279614111954831514248406805, 9.09506774958859620722915029988, 10.003299051398236020399911723529, 10.55776879274333003572184359291, 11.03289214573236192939953476523, 11.967259108523280208573786968315, 12.37891915346743271324374428236, 12.86155973523503006934451323458, 13.7133990453712065438744426516, 14.47936207670791532732614382855, 15.22457413032545126601002412891, 15.93360742971461689506852806757, 16.635236725918766388373658332427, 17.044175707109079781743765787069, 17.87485551758106976879751790327

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