Properties

Label 1-4033-4033.3623-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.899 + 0.436i$
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 + (−0.993 + 0.116i)3-s + (−0.5 + 0.866i)4-s + (−0.686 + 0.727i)5-s + (0.597 + 0.802i)6-s + (0.973 + 0.230i)7-s + 8-s + (0.973 − 0.230i)9-s + (0.973 + 0.230i)10-s + (0.973 − 0.230i)11-s + (0.396 − 0.918i)12-s + (−0.686 + 0.727i)13-s + (−0.286 − 0.957i)14-s + (0.597 − 0.802i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.993 + 0.116i)3-s + (−0.5 + 0.866i)4-s + (−0.686 + 0.727i)5-s + (0.597 + 0.802i)6-s + (0.973 + 0.230i)7-s + 8-s + (0.973 − 0.230i)9-s + (0.973 + 0.230i)10-s + (0.973 − 0.230i)11-s + (0.396 − 0.918i)12-s + (−0.686 + 0.727i)13-s + (−0.286 − 0.957i)14-s + (0.597 − 0.802i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.899 + 0.436i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.899 + 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9167606310 + 0.2106638022i\)
\(L(\frac12)\) \(\approx\) \(0.9167606310 + 0.2106638022i\)
\(L(1)\) \(\approx\) \(0.6592769660 - 0.03748792954i\)
\(L(1)\) \(\approx\) \(0.6592769660 - 0.03748792954i\)

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 \( 1 + (-0.993 + 0.116i)T \)
5 \( 1 + (-0.686 + 0.727i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (0.973 - 0.230i)T \)
13 \( 1 + (-0.686 + 0.727i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.286 - 0.957i)T \)
31 \( 1 + (0.893 + 0.448i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.597 - 0.802i)T \)
53 \( 1 + (0.597 + 0.802i)T \)
59 \( 1 + (0.597 - 0.802i)T \)
61 \( 1 + (0.893 - 0.448i)T \)
67 \( 1 + (0.396 + 0.918i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (-0.993 - 0.116i)T \)
83 \( 1 + (0.893 - 0.448i)T \)
89 \( 1 + (0.973 + 0.230i)T \)
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

−18.144806497547474625665343828820, −17.58905554524318064208154742163, −17.06611715170724173316354885584, −16.46220093974765971548147834859, −15.9711428427389365134715306346, −15.11403397110243921192669285288, −14.60459063617373539802403399119, −13.74813149780023837832181184188, −12.909076892545724290892476027377, −12.03829854034103485939639632761, −11.65506314473376879794413692846, −10.78907562334005524400522891481, −10.13493108079199847374108634007, −9.2713260524640036912199195228, −8.6037358820560531600106302477, −7.69912305608943655133125203433, −7.35663432824309113330569647332, −6.5973596431394569912299838009, −5.637094913981037539753357216423, −4.96304984273579580376953665628, −4.61217361562263125457904440829, −3.78931276564153608717577936275, −2.070575296020005179408244475098, −1.07536564608075304734326661852, −0.61971189678804320909882230174, 0.7902653289753400010027820490, 1.62526811437016606269612514854, 2.45310673720254146078913302705, 3.555038461153402950909522281875, 4.24265246116734065231723017789, 4.72684118457010784867082256905, 5.77302697720332532591838039402, 6.818760108638643167321050370674, 7.27358398047180808206881815155, 8.120730823737219984092915147521, 8.90735823543880228387809126983, 9.745604676221758183975216717787, 10.345034767751981841275831675093, 11.34554533763455386898125063842, 11.49853447307313389032131299146, 11.8130844005890458024040455229, 12.73553814646416867620458154857, 13.6693879936576382708990550926, 14.42120701537055880776652298561, 15.208001429342767130649681306363, 15.88829378688168591119227028487, 16.78869897596377014771575122233, 17.45756427088211223261340731352, 17.683239275887110671919099479292, 18.58187565944001114790044598049

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