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Properties

Label 1-4033-4033.32-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0321 - 0.999i$
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  + (−0.939 − 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s − i·5-s + (−0.766 + 0.642i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (0.939 − 0.342i)12-s + (−0.5 + 0.866i)13-s + (−0.939 − 0.342i)14-s + (−0.866 − 0.5i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s − i·5-s + (−0.766 + 0.642i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (0.939 − 0.342i)12-s + (−0.5 + 0.866i)13-s + (−0.939 − 0.342i)14-s + (−0.866 − 0.5i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0321 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0321 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0321 - 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0321 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.045020477 - 1.011898826i\)
\(L(\frac12)\) \(\approx\) \(1.045020477 - 1.011898826i\)
\(L(1)\) \(\approx\) \(0.8148280883 - 0.4447888441i\)
\(L(1)\) \(\approx\) \(0.8148280883 - 0.4447888441i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - iT \)
7 \( 1 + T \)
11 \( 1 + (0.342 + 0.939i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.642 - 0.766i)T \)
31 \( 1 + (0.342 + 0.939i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (-0.342 - 0.939i)T \)
97 \( 1 + (-0.642 - 0.766i)T \)
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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

−18.54391566707070337700557031638, −18.04887443455935216199347386580, −17.20393904716315039349323699585, −16.66657409645771531160506040452, −15.90248971589920068076639539260, −15.16068662505652216978375003113, −14.76485830881540945443921168910, −14.210074865265534813013795951863, −13.60915310352553080640713047424, −12.16232826752609668944515735852, −11.28765594726657606835208708328, −10.8275854592877783019532803692, −10.481866722789144405469584955702, −9.51436479651991061540777818804, −8.999764962713215224258123957705, −8.07451862699707202552072381198, −7.784118414876464476136056802344, −6.912551215735184686374533941733, −5.9190344883458616202616867328, −5.37856875325996146208149132432, −4.40346204678458137635160627102, −3.41823766961842299561733548311, −2.62726939558384071719463221781, −2.08223208112816114319786715278, −0.765272649863733763011930405616, 0.70163109115224623935750576786, 1.636811768027500032665550071080, 1.89655607973187231307443599939, 2.76912233126538494040231746385, 4.06547075776385474569450879302, 4.56919960513408519023711085836, 5.72160465344298798893121338841, 6.70929073803911426348476216199, 7.39368811043178369812397540859, 7.81259121803223049903079610385, 8.75942854793138326893674097682, 9.09355391942459478576945665487, 9.63415414002454317316717564223, 10.84619483933482863512039851390, 11.55293947925328405180776628660, 12.06399238130467409213080078372, 12.65661264542429588481213001011, 13.37494552221055157459917116475, 14.10773522616817814982922944244, 15.130655965379373290311551044783, 15.45187994421876644578226416224, 16.60389633735944876148933790124, 17.39096668768475652857187240863, 17.5195125240839027457289988016, 18.13871573284664447473102658815

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