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Properties

Label 1-4033-4033.1422-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.997 + 0.0698i$
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  + (0.173 + 0.984i)2-s + (−0.993 + 0.116i)3-s + (−0.939 + 0.342i)4-s + (0.973 + 0.230i)5-s + (−0.286 − 0.957i)6-s + (−0.286 − 0.957i)7-s + (−0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (0.893 − 0.448i)12-s + (−0.686 + 0.727i)13-s + (0.893 − 0.448i)14-s + (−0.993 − 0.116i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.993 + 0.116i)3-s + (−0.939 + 0.342i)4-s + (0.973 + 0.230i)5-s + (−0.286 − 0.957i)6-s + (−0.286 − 0.957i)7-s + (−0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (0.893 − 0.448i)12-s + (−0.686 + 0.727i)13-s + (0.893 − 0.448i)14-s + (−0.993 − 0.116i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.997 + 0.0698i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.997 + 0.0698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.137971798 + 0.03977184086i\)
\(L(\frac12)\) \(\approx\) \(1.137971798 + 0.03977184086i\)
\(L(1)\) \(\approx\) \(0.8072939337 + 0.2995259953i\)
\(L(1)\) \(\approx\) \(0.8072939337 + 0.2995259953i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (-0.993 + 0.116i)T \)
5 \( 1 + (0.973 + 0.230i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (-0.0581 - 0.998i)T \)
13 \( 1 + (-0.686 + 0.727i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.0581 + 0.998i)T \)
31 \( 1 + (0.597 - 0.802i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.686 - 0.727i)T \)
53 \( 1 + (0.893 - 0.448i)T \)
59 \( 1 + (-0.993 - 0.116i)T \)
61 \( 1 + (-0.0581 + 0.998i)T \)
67 \( 1 + (-0.835 + 0.549i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (-0.0581 - 0.998i)T \)
83 \( 1 + (0.973 + 0.230i)T \)
89 \( 1 + (0.597 + 0.802i)T \)
97 \( 1 + (-0.993 + 0.116i)T \)
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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

−18.42921873176779612662320675251, −17.75572516964870492937021650582, −17.44690524070541684719277885578, −16.696435561105145761657212585, −15.68067969305750116244672952053, −15.03206084172030483556989604133, −14.232348463317634236403113886972, −13.33472602506550817161101937456, −12.75315089484770664049037603295, −12.28816256262239244724933061875, −11.82058188410457107898742715688, −10.86344625555119878949644743868, −10.17230511726921847415489041740, −9.64168014730167432199952637971, −9.22675790148151521874732313282, −8.08565061351092850560457044092, −7.1519593826515796520248153730, −6.09283766416358225548165950988, −5.66688628718961893104956398776, −4.92346856942743226008708768747, −4.51382765605877702269526692213, −3.135105703928610309133014955807, −2.40401937568450905176844364392, −1.75092555077409051635983547480, −0.849914400561401153010872853, 0.457827165092910650243125792500, 1.34808241933156876456041888045, 2.7055710216171043039942187276, 3.74826054481904237057235785964, 4.48579559665175628264635945719, 5.12492702312858491232874403091, 5.9648004661146472398165695904, 6.46457546081572687944562729104, 6.95778383196501549059259549169, 7.714856478939850470107621846805, 8.88713464105425864461268808136, 9.35177062779509095933695284141, 10.387377731937256139346271462523, 10.59519540872236140024955985220, 11.58469562106961209000568254333, 12.62762891213064433883460769507, 13.14948302818904247353117244036, 13.64574551808542933852618944019, 14.51814009681406760390148165575, 14.97694300515197435893099708642, 16.06662361869741269093063782271, 16.63204145893989954081658120754, 16.97291271816605939661631825037, 17.50782328746209061790948816408, 18.156204584535962984301150687959

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