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Properties

Label 1-4033-4033.942-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.674 - 0.738i$
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  − 2-s + (−0.597 + 0.802i)3-s + 4-s + (0.230 + 0.973i)5-s + (0.597 − 0.802i)6-s + (−0.286 + 0.957i)7-s − 8-s + (−0.286 − 0.957i)9-s + (−0.230 − 0.973i)10-s + (0.230 − 0.973i)11-s + (−0.597 + 0.802i)12-s + (−0.973 + 0.230i)13-s + (0.286 − 0.957i)14-s + (−0.918 − 0.396i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.597 + 0.802i)3-s + 4-s + (0.230 + 0.973i)5-s + (0.597 − 0.802i)6-s + (−0.286 + 0.957i)7-s − 8-s + (−0.286 − 0.957i)9-s + (−0.230 − 0.973i)10-s + (0.230 − 0.973i)11-s + (−0.597 + 0.802i)12-s + (−0.973 + 0.230i)13-s + (0.286 − 0.957i)14-s + (−0.918 − 0.396i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.674 - 0.738i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (942, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.674 - 0.738i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1060192739 - 0.04675699496i\)
\(L(\frac12)\) \(\approx\) \(0.1060192739 - 0.04675699496i\)
\(L(1)\) \(\approx\) \(0.4186971181 + 0.2267128365i\)
\(L(1)\) \(\approx\) \(0.4186971181 + 0.2267128365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (0.230 + 0.973i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (0.230 - 0.973i)T \)
13 \( 1 + (-0.973 + 0.230i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.957 - 0.286i)T \)
31 \( 1 + (0.448 + 0.893i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.802 - 0.597i)T \)
53 \( 1 + (-0.116 + 0.993i)T \)
59 \( 1 + (-0.396 + 0.918i)T \)
61 \( 1 + (-0.549 - 0.835i)T \)
67 \( 1 + (0.802 - 0.597i)T \)
71 \( 1 - T \)
73 \( 1 + (0.286 + 0.957i)T \)
79 \( 1 + (-0.396 + 0.918i)T \)
83 \( 1 + (0.0581 + 0.998i)T \)
89 \( 1 + (-0.727 + 0.686i)T \)
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

−18.41124603344713725626907085492, −17.691998594428804083679247997249, −17.20122017946624869469771624896, −16.90781265821492263638725970079, −16.13353845744662261253252829897, −15.497651478500735251472141937593, −14.448487518475637105381006389615, −13.52923253659704002726803879445, −13.03197921491885636044621823201, −12.14022446571476112566932039909, −11.82102994486084592447876312037, −11.009759694722557475655033391933, −9.9709646969025736989221245813, −9.73477231854574580307926315981, −8.91052102305044867939882038198, −7.86867192444795597131705642085, −7.32495679313088978901842906793, −7.04468079580084000300779114501, −5.983728728516089575126206268176, −5.24679061020813361811132535311, −4.53607757313707178515615513755, −3.31232517063126190969277336685, −2.17341193453370836728640621938, −1.59272991327123119978281260187, −0.7669093667208950496599245861, 0.065013336199157292065896152609, 1.45891792440689371325134903156, 2.619199454628344325939280780181, 2.98884281784296751011295339293, 3.91763758783626765910311398807, 5.18655242455157240700392285554, 5.8301092846910801017529036870, 6.49346868349900977252771096938, 6.96518372075781089738532168818, 8.11313851227696740602450882429, 8.85063535911822355528225192832, 9.48172948225201704749041612869, 10.0150161501969949554330577981, 10.80044319128091260295087652799, 11.23172042537432168132752327065, 11.941882696614920120512236117444, 12.50652768002838788405735050240, 13.79779460839412282160878713520, 14.7020919127031223366645107950, 15.16179108046341131686150814503, 15.69144857280189316486092858325, 16.506732514451433838938867052512, 17.01264948465623779019361016925, 17.69268219243794397011858111752, 18.42818231177726816647380652562

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