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Properties

Label 1-4033-4033.73-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.985 - 0.169i$
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  + (0.939 + 0.342i)2-s + (0.973 − 0.230i)3-s + (0.766 + 0.642i)4-s + (0.835 − 0.549i)5-s + (0.993 + 0.116i)6-s + (−0.0581 − 0.998i)7-s + (0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.893 + 0.448i)12-s + (0.0581 + 0.998i)13-s + (0.286 − 0.957i)14-s + (0.686 − 0.727i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.973 − 0.230i)3-s + (0.766 + 0.642i)4-s + (0.835 − 0.549i)5-s + (0.993 + 0.116i)6-s + (−0.0581 − 0.998i)7-s + (0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (0.893 + 0.448i)12-s + (0.0581 + 0.998i)13-s + (0.286 − 0.957i)14-s + (0.686 − 0.727i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.347655964 - 0.5413147084i\)
\(L(\frac12)\) \(\approx\) \(6.347655964 - 0.5413147084i\)
\(L(1)\) \(\approx\) \(3.133704664 + 0.01267402297i\)
\(L(1)\) \(\approx\) \(3.133704664 + 0.01267402297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (0.973 - 0.230i)T \)
5 \( 1 + (0.835 - 0.549i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (0.973 + 0.230i)T \)
13 \( 1 + (0.0581 + 0.998i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.686 + 0.727i)T \)
31 \( 1 + (0.0581 - 0.998i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.396 - 0.918i)T \)
53 \( 1 + (-0.835 + 0.549i)T \)
59 \( 1 + (0.686 - 0.727i)T \)
61 \( 1 + (-0.396 - 0.918i)T \)
67 \( 1 + (-0.0581 + 0.998i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.973 + 0.230i)T \)
79 \( 1 + (-0.893 + 0.448i)T \)
83 \( 1 + (-0.286 + 0.957i)T \)
89 \( 1 + (0.993 + 0.116i)T \)
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

−18.818254883621988461037229955553, −17.818259041978063355795340482970, −17.254595964300528804289406308668, −15.99336403400408925999834261330, −15.504454051451439575584832156115, −14.887007116051857435348481448422, −14.441834876485743561742733149619, −13.74064651071103670952325059903, −13.16270832739030096879449245613, −12.49378876699150573015805932889, −11.80331458978707593803070144311, −10.69768360583678118016655087940, −10.41360142445256109646120652036, −9.4974241108685904197803650256, −8.93075606789490258397320332295, −8.10229145042305912771382037209, −7.09428845606928745397263367113, −6.2619058598958983436814616822, −5.85872902491722169221084335351, −4.94148563089081222444053039583, −4.03948563446266892062422916427, −3.31528883712922125298631721068, −2.67749590447718872577496356284, −2.03868659173301262226165031069, −1.35701739768962180184924644186, 1.14902817109809295411746822310, 1.86457171565284193267008416914, 2.538485901144657215190042133839, 3.56538355890438273962227625, 4.40026798258818744510875913824, 4.50636908803371664779276894280, 5.85895775933131962236312325421, 6.73532572122161348827727827728, 6.86607355225525395592469072925, 7.96645904823553585631278541488, 8.57489378320657167532409351713, 9.43590258837103741166828280226, 9.98001438664911587060001612146, 10.9573255673922503730539028005, 11.92672222926209776527959294493, 12.51248466188715762653320442485, 13.22728831407778056680157396493, 13.79191456461251969816466621027, 14.36843791131131502586392575710, 14.51398434232541887656284503765, 15.73065725154097695304336565544, 16.414115688818899228690735320061, 16.869238533116686512605958381803, 17.57210786925159114091044436918, 18.4556197580196330456183010994

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