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Properties

Label 1-4033-4033.192-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.693 - 0.720i$
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  + (0.5 − 0.866i)2-s + (0.396 − 0.918i)3-s + (−0.5 − 0.866i)4-s + (−0.286 − 0.957i)5-s + (−0.597 − 0.802i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (−0.973 − 0.230i)10-s + (−0.973 + 0.230i)11-s + (−0.993 + 0.116i)12-s + (0.286 + 0.957i)13-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.396 − 0.918i)3-s + (−0.5 − 0.866i)4-s + (−0.286 − 0.957i)5-s + (−0.597 − 0.802i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (−0.973 − 0.230i)10-s + (−0.973 + 0.230i)11-s + (−0.993 + 0.116i)12-s + (0.286 + 0.957i)13-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.693 - 0.720i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.693 - 0.720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5392647486 - 0.2294771047i\)
\(L(\frac12)\) \(\approx\) \(0.5392647486 - 0.2294771047i\)
\(L(1)\) \(\approx\) \(0.5856762008 - 0.7207496750i\)
\(L(1)\) \(\approx\) \(0.5856762008 - 0.7207496750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (-0.286 - 0.957i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (-0.973 + 0.230i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.286 - 0.957i)T \)
31 \( 1 + (0.893 + 0.448i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.597 + 0.802i)T \)
53 \( 1 + (-0.396 + 0.918i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (-0.0581 + 0.998i)T \)
67 \( 1 + (0.993 + 0.116i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (-0.597 + 0.802i)T \)
83 \( 1 + (-0.835 - 0.549i)T \)
89 \( 1 + (-0.286 - 0.957i)T \)
97 \( 1 + (0.893 - 0.448i)T \)
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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

−18.41629397220693671103021090615, −17.732777154922026314900217556510, −17.0323776155269597500269257885, −16.08363368393640391457858747622, −15.87925298171855617683469552725, −15.20540131192206544387949680980, −14.64421154827144759574772381252, −13.8806418124343745262544416028, −13.328074980947603329666099522005, −12.78109441539247321031154930745, −11.535042204994872579514144388117, −10.907657125843516252119081000493, −10.14084841707842050714842522880, −9.72618982976525013194025973431, −8.555217712798538573284133462336, −8.01276999552913153740093743331, −7.41103847902417363606982927796, −6.566139154816280995341063241223, −5.84364794994203234460879869809, −5.12639821356470119289786160594, −4.21247758835187381382647736299, −3.461828808953295574885259021417, −3.215827480383532438709298586304, −2.27691450353129448798592081901, −0.170215969483492364765889424248, 0.72936385860826232918626112938, 1.81064721444432276579370751193, 2.44657442480998527520628573409, 2.98150425354129330698630046448, 4.17805439915863541766280405864, 4.66184887829514995968787065237, 5.65011824512865577429864558021, 6.312744874741637314675634584688, 7.022043466165693895512639639583, 8.22460440356758130643805977884, 8.74068410984547635957732330543, 9.26455748230818445610444874724, 10.00971153207298691766932797230, 11.17429140253805190260280571352, 11.67702554298115684601017301988, 12.37631114323044301774173797448, 12.96298538334161788109979132353, 13.26920316872508575707889356592, 13.95457103504337218643511231509, 14.91058838696561196940336724858, 15.6020390285850453448503152741, 16.0910041977437003778886496183, 17.294014903413210585610307960615, 17.91243761614016823351589592148, 18.71295347601565277372297283500

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