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Properties

Label 1-4033-4033.857-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.598 - 0.800i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.342 − 0.939i)2-s + (0.686 + 0.727i)3-s + (−0.766 − 0.642i)4-s + (−0.448 + 0.893i)5-s + (0.918 − 0.396i)6-s + (−0.835 + 0.549i)7-s + (−0.866 + 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.686 + 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.0581 − 0.998i)12-s + (0.549 + 0.835i)13-s + (0.230 + 0.973i)14-s + (−0.957 + 0.286i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.686 + 0.727i)3-s + (−0.766 − 0.642i)4-s + (−0.448 + 0.893i)5-s + (0.918 − 0.396i)6-s + (−0.835 + 0.549i)7-s + (−0.866 + 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.686 + 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.0581 − 0.998i)12-s + (0.549 + 0.835i)13-s + (0.230 + 0.973i)14-s + (−0.957 + 0.286i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.598 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.598 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.598 - 0.800i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.598 - 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.084138415 - 0.5430696730i\)
\(L(\frac12)\) \(\approx\) \(1.084138415 - 0.5430696730i\)
\(L(1)\) \(\approx\) \(1.054974844 + 0.02833284023i\)
\(L(1)\) \(\approx\) \(1.054974844 + 0.02833284023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
3 \( 1 + (0.686 + 0.727i)T \)
5 \( 1 + (-0.448 + 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.686 + 0.727i)T \)
13 \( 1 + (0.549 + 0.835i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.957 + 0.286i)T \)
31 \( 1 + (-0.549 + 0.835i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.597 - 0.802i)T \)
53 \( 1 + (-0.893 - 0.448i)T \)
59 \( 1 + (-0.957 + 0.286i)T \)
61 \( 1 + (-0.802 - 0.597i)T \)
67 \( 1 + (-0.835 - 0.549i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.686 - 0.727i)T \)
79 \( 1 + (0.998 + 0.0581i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (0.918 - 0.396i)T \)
97 \( 1 + (0.998 - 0.0581i)T \)
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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

−18.43187444695441084183519940529, −17.56328530967920771117226480858, −16.90519433100132735490146680773, −16.294551380292253002035755498386, −15.67597770934131812633203378038, −15.10010297884175404967146792986, −14.22408279323438541912616242533, −13.49045799131512109985677346692, −13.11944907628909165647082488561, −12.57674445588256388689417474406, −12.00047466579546786646625643204, −10.787026993681734916800816267090, −9.79666728384075422016820504455, −9.09917725382769609615305429463, −8.39877335324767057512424482214, −7.676992917877821274756140333225, −7.618665732578771301687610058052, −6.26038904929369312626906628249, −5.96833767039586245881847788188, −5.046771709436702325709771029, −3.98672205085903601820250282559, −3.42925599546308915078557600546, −2.88096247825774831543640923783, −1.288195662084985867173372247405, −0.62002711198819252384313104683, 0.2142110779346821294349338655, 1.762932418632630291092264088640, 2.50398411424358814938171327572, 3.14820386802098930021635256029, 3.5416727998031494222811108509, 4.584627262039445549946283324834, 5.02019099556970713957041408948, 6.19199408526063835434525106800, 6.87189531726098433524767242200, 7.9078515476413629091272117172, 8.79020702369377095623841154324, 9.259362374048784855509945384214, 10.15373758023536681480349711386, 10.48490551633299654706338705768, 11.18039770709290040540282523772, 12.05944863754463318590015637572, 12.649495129516074978555799066500, 13.48521900939215557034947912156, 14.078390051210091739225612787120, 14.86757608156619635827175103363, 15.20346557946714184745483298556, 15.9500830820341880002677521365, 16.63121053388956530321307977306, 17.87610111341991356336145422095, 18.569861903778868289038648797053

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