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Properties

Label 1-4033-4033.1380-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.617 + 0.786i$
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 + (0.893 + 0.448i)3-s + (0.939 − 0.342i)4-s + (−0.396 − 0.918i)5-s + (0.957 + 0.286i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.549 − 0.835i)10-s + (−0.549 + 0.835i)11-s + (0.993 + 0.116i)12-s + (0.116 − 0.993i)13-s + (−0.998 + 0.0581i)14-s + (0.0581 − 0.998i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.893 + 0.448i)3-s + (0.939 − 0.342i)4-s + (−0.396 − 0.918i)5-s + (0.957 + 0.286i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.549 − 0.835i)10-s + (−0.549 + 0.835i)11-s + (0.993 + 0.116i)12-s + (0.116 − 0.993i)13-s + (−0.998 + 0.0581i)14-s + (0.0581 − 0.998i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.617 + 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.828771614 + 1.862817372i\)
\(L(\frac12)\) \(\approx\) \(3.828771614 + 1.862817372i\)
\(L(1)\) \(\approx\) \(2.096481366 - 0.08658057169i\)
\(L(1)\) \(\approx\) \(2.096481366 - 0.08658057169i\)

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 \( 1 + (0.893 + 0.448i)T \)
5 \( 1 + (-0.396 - 0.918i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (-0.549 + 0.835i)T \)
13 \( 1 + (0.116 - 0.993i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.893 + 0.448i)T \)
31 \( 1 + (0.597 + 0.802i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.230 + 0.973i)T \)
53 \( 1 + (0.802 + 0.597i)T \)
59 \( 1 + (-0.998 - 0.0581i)T \)
61 \( 1 + (-0.686 - 0.727i)T \)
67 \( 1 + (-0.918 - 0.396i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.835 - 0.549i)T \)
79 \( 1 + (-0.116 - 0.993i)T \)
83 \( 1 + (-0.893 - 0.448i)T \)
89 \( 1 + (0.686 + 0.727i)T \)
97 \( 1 + (-0.396 - 0.918i)T \)
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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

−18.41157452043323098538631825946, −17.60200431282488151895803783470, −16.49092692502319695637102504857, −15.76905439710218264247647586816, −15.50201073958151962572048574024, −14.73806009412023861076935764540, −13.83485823161926202303137381400, −13.67447178819367099489415980280, −13.01001108775399290272493593504, −12.03801465243350634413914082086, −11.64914973593374783477658847748, −10.68376586274949167124123802693, −10.025807849178374300970311732019, −9.03535258199096586034093988651, −8.24062613802404814774025396449, −7.54658035240438335249969353108, −6.81741965149026207039782321183, −6.35730530558160200754787545349, −5.72653746781484702441391330305, −4.22704804690077021958648588533, −3.929255663375695230767706366411, −2.93092982564519060577449856058, −2.68511125509845793409158514980, −1.78837956876770308905228288978, −0.37377047186814225251776038524, 0.88978293386301516683436706506, 1.85574883625694815582458027315, 2.8974277559118099104246552345, 3.200052227605379975438216906196, 4.24484049502677516017940475870, 4.64200273666848906829085865646, 5.428103287384189173280827118451, 6.27906633776754800472381115381, 7.382359102513440796985561400981, 7.73599765112388294889958780630, 8.6731526536887767665948015822, 9.55021048015752682916353916502, 10.140242720051459646926988850252, 10.64814223052422570487295931693, 11.871663252224305282733926304567, 12.47496746728076416688135051794, 12.91578582923556609587266228822, 13.63739184997571220602105011341, 14.1216388656376464480415666407, 15.13088496907714558603789737231, 15.725063225934914626014135883995, 15.96697651155646632337736217429, 16.56111338597503187452370274147, 17.71901460997019121244938703987, 18.61955470942337305882051970407

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