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Properties

Label 1-4033-4033.146-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.543 - 0.839i$
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 + (0.686 + 0.727i)3-s + (−0.5 + 0.866i)4-s + (−0.549 − 0.835i)5-s + (−0.286 + 0.957i)6-s + (−0.0581 − 0.998i)7-s − 8-s + (−0.0581 + 0.998i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (−0.973 + 0.230i)12-s + (0.835 − 0.549i)13-s + (0.835 − 0.549i)14-s + (0.230 − 0.973i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.686 + 0.727i)3-s + (−0.5 + 0.866i)4-s + (−0.549 − 0.835i)5-s + (−0.286 + 0.957i)6-s + (−0.0581 − 0.998i)7-s − 8-s + (−0.0581 + 0.998i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (−0.973 + 0.230i)12-s + (0.835 − 0.549i)13-s + (0.835 − 0.549i)14-s + (0.230 − 0.973i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.543 - 0.839i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.543 - 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.071245799 - 0.5824432642i\)
\(L(\frac12)\) \(\approx\) \(1.071245799 - 0.5824432642i\)
\(L(1)\) \(\approx\) \(1.167629455 + 0.4159546437i\)
\(L(1)\) \(\approx\) \(1.167629455 + 0.4159546437i\)

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 \( 1 + (0.686 + 0.727i)T \)
5 \( 1 + (-0.549 - 0.835i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (-0.448 - 0.893i)T \)
13 \( 1 + (0.835 - 0.549i)T \)
17 \( 1 + T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.549 - 0.835i)T \)
31 \( 1 + (-0.802 + 0.597i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (-0.957 + 0.286i)T \)
53 \( 1 + (-0.727 + 0.686i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (0.116 - 0.993i)T \)
67 \( 1 + (0.230 - 0.973i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.835 + 0.549i)T \)
79 \( 1 + (0.286 + 0.957i)T \)
83 \( 1 + (-0.396 - 0.918i)T \)
89 \( 1 + (0.549 + 0.835i)T \)
97 \( 1 + (0.802 + 0.597i)T \)
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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

−18.57974449782764553477172701848, −18.29274702715371543063292535219, −17.856989993626176394927766859652, −16.24461392405337789521719472611, −15.61384980257811180292196541662, −14.774693543107270706106919802451, −14.529009223511922809650507002511, −13.840224512186806509693881050713, −12.92655669238737727957157633633, −12.440307238155843447489927301832, −11.771848065706418859793145360924, −11.346814097640758851674385902408, −10.28132023542312732057461060847, −9.65222943570294299375485906267, −8.97555913133596203847439323243, −8.08836782026952048879465568623, −7.48206554096787916595959592805, −6.52960693377617599701139417186, −5.913294018315623162833631106661, −5.07720669997872161186639829478, −3.87249361946278697165487079328, −3.4593794618563187900116027427, −2.65890871703155295524444877231, −2.02494719950426571595122049810, −1.31180950510135627398292166187, 0.25484417179043196111889109939, 1.41895448404129028285089175909, 3.03815839212806383878369922421, 3.55082625418513612010797878076, 3.96879959759162906360429462029, 4.95772985793210735222699024608, 5.39775660425454060931578571006, 6.26088750356877555081913059897, 7.5320177867643288507060878366, 7.90881452407386588573453383810, 8.28793555180508399737232407280, 9.27677401458912245091188713373, 9.77412028231986170477986646475, 10.85568718731183221663278970142, 11.4362561515983235317192058717, 12.49760252496313154505883157960, 13.16610994649980432229475614321, 13.84807281079353543340652824088, 14.056210955170558137529255302381, 15.167512487516580418389530559515, 15.684419387413745658889229273917, 16.37127487494293110442366875618, 16.4876531494225719483511347871, 17.36818431324063636160538882842, 18.298803395706391399543644825165

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