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Properties

Label 1-4033-4033.1010-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.937 + 0.347i$
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  + (0.173 − 0.984i)2-s + (0.973 + 0.230i)3-s + (−0.939 − 0.342i)4-s + (0.835 + 0.549i)5-s + (0.396 − 0.918i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 + 0.866i)8-s + (0.893 + 0.448i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (−0.835 − 0.549i)12-s + (−0.0581 + 0.998i)13-s + (0.973 + 0.230i)14-s + (0.686 + 0.727i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.973 + 0.230i)3-s + (−0.939 − 0.342i)4-s + (0.835 + 0.549i)5-s + (0.396 − 0.918i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 + 0.866i)8-s + (0.893 + 0.448i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (−0.835 − 0.549i)12-s + (−0.0581 + 0.998i)13-s + (0.973 + 0.230i)14-s + (0.686 + 0.727i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1010, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.937 + 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.512040480 + 0.6300230088i\)
\(L(\frac12)\) \(\approx\) \(3.512040480 + 0.6300230088i\)
\(L(1)\) \(\approx\) \(1.890704325 - 0.1263307276i\)
\(L(1)\) \(\approx\) \(1.890704325 - 0.1263307276i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (0.835 + 0.549i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (0.686 + 0.727i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.286 + 0.957i)T \)
31 \( 1 + (0.835 - 0.549i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.597 + 0.802i)T \)
53 \( 1 + (0.0581 - 0.998i)T \)
59 \( 1 + (-0.686 - 0.727i)T \)
61 \( 1 + (-0.396 + 0.918i)T \)
67 \( 1 + (-0.893 - 0.448i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.686 - 0.727i)T \)
79 \( 1 + (-0.835 + 0.549i)T \)
83 \( 1 + (-0.686 + 0.727i)T \)
89 \( 1 + (-0.597 - 0.802i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
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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

−18.32186946829507817114360422257, −17.533978503187410426189542122130, −17.07724195087954536535186981097, −16.34901950565656715529989222635, −15.80660303942302723427785213943, −14.807628391754263323057142197930, −14.28976997271954338574725260467, −13.69865123286622139009215069375, −13.36620158050953303141572129333, −12.63056569213853185909762067384, −11.88559142026681912512036163646, −10.45138209826320541432337444141, −9.86174688450287510177345288174, −9.309842629053954780365362391090, −8.56602420750511422067659732486, −7.83642338368961621816176927493, −7.4422870867329561378033770733, −6.41612436381152860578794305312, −5.915353320798543274178686959068, −4.97650255475368496019519964203, −4.22670426292714677725162296377, −3.38777752411819362675625644513, −2.83457640571401745318801539247, −1.217038126209340001344767569457, −0.96158069008581958707677082855, 1.38642625376850475047726739764, 1.88080894068923185753642396206, 2.70680019918858685336750778858, 3.06762800503176110144328089796, 4.1640895435364448525949813199, 4.74949483524651607067946528246, 5.64849852377697766901754874948, 6.54103886007339832823967607294, 7.29630048750161813748091309867, 8.529671236674030369623349005941, 9.012156762393061156721855312845, 9.52986084457046083500117757823, 10.064315599669086485953327599078, 10.887789035263804603455589950280, 11.61828613367390213461877615712, 12.56883101576998476796573501436, 12.85364212004191836908951384700, 13.93750738404622933152111418, 14.24550007433890494229709443140, 14.88768270115947483854150709619, 15.38580307281810375732623339463, 16.537520839364285750589756207628, 17.43520928983346434694796407515, 18.0142668973689603327725364040, 18.88262315078178260301071979053

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