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Properties

Label 1-4033-4033.3400-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.696 - 0.717i$
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 + (−0.0581 − 0.998i)3-s + (0.173 + 0.984i)4-s + (−0.993 − 0.116i)5-s + (0.597 − 0.802i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (0.973 − 0.230i)12-s + (0.396 + 0.918i)13-s + (0.973 − 0.230i)14-s + (−0.0581 + 0.998i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.0581 − 0.998i)3-s + (0.173 + 0.984i)4-s + (−0.993 − 0.116i)5-s + (0.597 − 0.802i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (0.973 − 0.230i)12-s + (0.396 + 0.918i)13-s + (0.973 − 0.230i)14-s + (−0.0581 + 0.998i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.696 - 0.717i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07633656821 - 0.1804761191i\)
\(L(\frac12)\) \(\approx\) \(0.07633656821 - 0.1804761191i\)
\(L(1)\) \(\approx\) \(1.043001603 + 0.1612851140i\)
\(L(1)\) \(\approx\) \(1.043001603 + 0.1612851140i\)

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

−18.67590647376870242080503681517, −18.4081554755882164573621248215, −17.375706208015128400620833329994, −16.191979140225820407530563716585, −15.84303057503903693101181928257, −15.30653171330294368752528554755, −14.7130185852917324645338847813, −14.109047148860936408232502240694, −13.21685742287914733306288412940, −12.377462519463015937195606540722, −11.70152645783211603666357183957, −11.28478564426380170094726338660, −10.57097846325974994723120719922, −10.119198981175829082523643228, −8.9742005900557703550718587176, −8.4668241060353512449405106635, −7.713083835980356287721275886805, −6.49941210898186497301209204921, −5.588795544012460182774626358, −5.18130071946990044564236165116, −4.48978539599331934016944358862, −3.65644531379933919371157169613, −2.97489704419640366680002820, −2.54319958288884259647194203049, −1.06060045442637715052946597524, 0.0439794970318485228257612340, 1.52797265747432049054270961119, 2.15519305778677030324856752282, 3.40370068014058950304649690622, 3.989154734695829795633410456620, 4.66085454514494329870721287931, 5.543864122793469164494562075851, 6.34326677861110718907625579766, 7.06395106684785659848461642004, 7.70018075707771956526978588036, 8.06880868090658011431632733320, 8.630733508620724887957904783542, 10.04042113302332547159042319403, 10.93415111837119648066744214884, 11.797245685818431261779282639499, 12.011979797504390968304122274091, 12.79717702055345889001001445722, 13.58804396284401668630242731268, 13.947230360392257247717091558, 14.994871136552541007731306922371, 15.11620020993488618620229909041, 16.34058714427614755827879866882, 16.79600261423107136366054518025, 17.29383025171130331896536725721, 18.30523245752814046485489036081

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