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Properties

Label 1-4033-4033.39-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.328 - 0.944i$
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 + (−0.727 + 0.686i)5-s + (0.597 − 0.802i)6-s + (0.973 − 0.230i)7-s − 8-s + (0.973 + 0.230i)9-s + (0.230 + 0.973i)10-s + (−0.230 + 0.973i)11-s + (−0.396 − 0.918i)12-s + (0.686 + 0.727i)13-s + (0.286 − 0.957i)14-s + (−0.802 + 0.597i)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 + (−0.727 + 0.686i)5-s + (0.597 − 0.802i)6-s + (0.973 − 0.230i)7-s − 8-s + (0.973 + 0.230i)9-s + (0.230 + 0.973i)10-s + (−0.230 + 0.973i)11-s + (−0.396 − 0.918i)12-s + (0.686 + 0.727i)13-s + (0.286 − 0.957i)14-s + (−0.802 + 0.597i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.518806818 - 1.790697636i\)
\(L(\frac12)\) \(\approx\) \(2.518806818 - 1.790697636i\)
\(L(1)\) \(\approx\) \(1.633005200 - 0.6863618189i\)
\(L(1)\) \(\approx\) \(1.633005200 - 0.6863618189i\)

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 \( 1 + (-0.727 + 0.686i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (-0.230 + 0.973i)T \)
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 \( 1 + (0.957 + 0.286i)T \)
31 \( 1 + (-0.448 - 0.893i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.802 + 0.597i)T \)
53 \( 1 + (0.802 + 0.597i)T \)
59 \( 1 + (-0.597 - 0.802i)T \)
61 \( 1 + (0.448 - 0.893i)T \)
67 \( 1 + (0.918 + 0.396i)T \)
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 \( 1 + (0.230 + 0.973i)T \)
97 \( 1 + (0.448 - 0.893i)T \)
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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

−18.400869760248746573408761426541, −17.99314392290785349601228557632, −17.09008807966071939605416161246, −16.35029914350048707932217952601, −15.68343899822868834968729331206, −15.18461055285281345356528943689, −14.6816324739741342668756595955, −13.858559232390871216847688151870, −13.24439275987613021307650682707, −12.7592222968444680608278714511, −11.96142402576869341159269273003, −11.19022242541196443607150890118, −10.298185101855261145519198464850, −9.01925974893120512064449518615, −8.593266387046699244087521017443, −8.06975346803576843360467293696, −7.81029468890797012123349845894, −6.70177548723532778656016533668, −5.87471793109169345541067175453, −5.074848989266955441110335431208, −4.3678811985785422754358517070, −3.62562306525539154208001867622, −3.13340593415325769767534082167, −1.895871558774754866102919674351, −0.95291999609406756461831400718, 0.75364223329017434669217416876, 2.04506464888890802370330663196, 2.27270517193318068435330945874, 3.20320163979558755721768930098, 4.19676060543869124917587372803, 4.35490790260923301045853051341, 5.149558435114587116718261126013, 6.6754319274662834709827149580, 6.99364376043124212133593962202, 8.05252298854377291992905552158, 8.65440367934864267998774492339, 9.35317458286743464551382145814, 10.25743186792780964141249557770, 10.82755888779354238818728712319, 11.37452547936492816781588059827, 12.15570094295926283101105471052, 12.89057672450204913367588546936, 13.68154584645547277145205686889, 14.30363606265185696328094928779, 14.64993340932856635487991542182, 15.514756283926636691834923314382, 15.73872434717500941457810139579, 17.13801828948105702603953919022, 18.11148272976398993622487164966, 18.51237768222627515707534006331

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