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Properties

Label 1-4033-4033.2388-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.635 + 0.772i$
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  + (−0.766 + 0.642i)2-s + (0.835 + 0.549i)3-s + (0.173 − 0.984i)4-s + (−0.918 − 0.396i)5-s + (−0.993 + 0.116i)6-s + (−0.993 + 0.116i)7-s + (0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.957 − 0.286i)10-s + (−0.957 − 0.286i)11-s + (0.686 − 0.727i)12-s + (−0.597 − 0.802i)13-s + (0.686 − 0.727i)14-s + (−0.549 − 0.835i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.835 + 0.549i)3-s + (0.173 − 0.984i)4-s + (−0.918 − 0.396i)5-s + (−0.993 + 0.116i)6-s + (−0.993 + 0.116i)7-s + (0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.957 − 0.286i)10-s + (−0.957 − 0.286i)11-s + (0.686 − 0.727i)12-s + (−0.597 − 0.802i)13-s + (0.686 − 0.727i)14-s + (−0.549 − 0.835i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.635 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5490991596 + 0.2593558371i\)
\(L(\frac12)\) \(\approx\) \(0.5490991596 + 0.2593558371i\)
\(L(1)\) \(\approx\) \(0.5765333636 + 0.2026405899i\)
\(L(1)\) \(\approx\) \(0.5765333636 + 0.2026405899i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (0.835 + 0.549i)T \)
5 \( 1 + (-0.918 - 0.396i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (-0.957 - 0.286i)T \)
13 \( 1 + (-0.597 - 0.802i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.957 + 0.286i)T \)
31 \( 1 + (-0.998 + 0.0581i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.802 + 0.597i)T \)
53 \( 1 + (-0.727 - 0.686i)T \)
59 \( 1 + (0.835 - 0.549i)T \)
61 \( 1 + (-0.957 + 0.286i)T \)
67 \( 1 + (0.230 + 0.973i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.973 + 0.230i)T \)
79 \( 1 + (0.286 - 0.957i)T \)
83 \( 1 + (-0.396 + 0.918i)T \)
89 \( 1 + (-0.998 - 0.0581i)T \)
97 \( 1 + (-0.549 + 0.835i)T \)
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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

−18.60151926766720955962237678640, −18.07883194092729245110768004516, −17.092572870423240065365681161599, −16.3510612674739313822422508589, −15.67023172271330947115714575654, −15.1456716684419572579225006102, −14.18350839032172366703345931347, −13.4026725797927537002139863295, −12.7404805701936257853936915400, −12.22878850841384136214266460402, −11.59549495505310125203971149516, −10.68217273523192546790100246906, −9.947131752231036889266028062352, −9.41460434970360037770834145233, −8.67555694317908093366906273312, −7.83487007298103948082296318465, −7.33577886409960487691315905334, −6.979560600991101159234288212814, −5.89109739200573756062358806482, −4.425991410447243174461415917575, −3.70594067689448867918292639436, −3.14841847859659677678375006214, −2.45310407104601997327093391162, −1.6965128476271986905287275214, −0.431829493355859638147960100271, 0.43425734413873123547034951992, 1.74725773390417521792745892776, 2.85548141573833792000083826853, 3.36362787473826221882302107230, 4.33110295034458393430191586530, 5.36327515667807019287344522741, 5.64138690473992186189753041419, 7.03386222439285253962353341207, 7.65493793999130031092668538226, 8.00994387206799968621746670828, 8.78454096611394555761854871263, 9.515613885474099604559947222104, 10.08464031173661705630751461418, 10.58820507021485682642502805038, 11.58308639968578674390905238193, 12.52831818688164791293704127736, 13.14183281902638207844034123225, 14.059213545378520604574204412716, 14.72626603633727938372784532193, 15.38317130514889047639673312164, 15.9193093797071967376896221181, 16.41678983257310176063653596134, 16.755767410074224524852853679951, 18.110545994814818485211024767196, 18.66251795209190095099331085508

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