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Properties

Label 1-4033-4033.584-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.940 - 0.338i$
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 + (−0.973 + 0.230i)3-s + (0.173 − 0.984i)4-s + (0.549 + 0.835i)5-s + (0.597 − 0.802i)6-s + (−0.0581 − 0.998i)7-s + (0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.957 − 0.286i)10-s + (0.957 − 0.286i)11-s + (0.0581 + 0.998i)12-s + (0.0581 + 0.998i)13-s + (0.686 + 0.727i)14-s + (−0.727 − 0.686i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.973 + 0.230i)3-s + (0.173 − 0.984i)4-s + (0.549 + 0.835i)5-s + (0.597 − 0.802i)6-s + (−0.0581 − 0.998i)7-s + (0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.957 − 0.286i)10-s + (0.957 − 0.286i)11-s + (0.0581 + 0.998i)12-s + (0.0581 + 0.998i)13-s + (0.686 + 0.727i)14-s + (−0.727 − 0.686i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.940 - 0.338i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.940 - 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8118281220 - 0.1417202961i\)
\(L(\frac12)\) \(\approx\) \(0.8118281220 - 0.1417202961i\)
\(L(1)\) \(\approx\) \(0.6281634692 + 0.1521648522i\)
\(L(1)\) \(\approx\) \(0.6281634692 + 0.1521648522i\)

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

−18.146389015216864104282433001139, −17.87543048882704433614390680602, −17.45183053024211846230275891101, −16.56849410669571164676667358946, −16.1406022962763126639865280734, −15.43024031054185548287193746288, −14.40334112357940442738153274260, −13.16114073847949428151303520321, −12.779715130584965897061307809747, −12.40179270322882393763550012732, −11.48925664289191203380020868178, −11.13634271028715958262630589690, −10.13123297973974924836483055345, −9.57642685954603722672864093439, −8.84318623516292296461408740029, −8.37659830955905986152713793335, −7.27571413068131694710053816990, −6.59046801913993693594255900267, −5.8244490165049654862775625547, −5.01940156925429333291374588884, −4.41384656145694538391820309071, −3.21704399532705164394257228140, −2.33714394116009002468666609250, −1.430485667425499071730793841508, −0.96357410953836232939667822454, 0.4263818462201254174783731471, 1.39338275545731360625616098708, 2.118918469714316909209973649748, 3.60541889591318054308210446349, 4.24044365312531071168573341693, 5.24347526153431306662891621534, 5.93710818856330749345340827727, 6.68444084118409172604566865533, 7.00179317797320914566616883437, 7.61864653300562702051746535739, 8.97907417557690921331759339297, 9.53394676637712061015749575693, 10.00554422358105733432481767824, 10.89900462117598777568951306859, 11.281473217319277543089229831961, 11.81194240475332671271969483707, 13.26254526449060374570771263568, 13.83142363965586510210302126139, 14.36391055424731062890819094959, 15.246523438364842185987879900720, 15.86381974835848261862437594540, 16.6484640873775225102960196210, 17.205706779701418449159871871451, 17.41945871293869091759688499994, 18.3441711933005387058722818534

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