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Properties

Label 1-4033-4033.471-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.429 - 0.903i$
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 + (0.396 + 0.918i)3-s + (0.173 − 0.984i)4-s + (−0.973 + 0.230i)5-s + (−0.893 − 0.448i)6-s + (−0.286 + 0.957i)7-s + (0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (0.597 − 0.802i)10-s + (0.597 + 0.802i)11-s + (0.973 − 0.230i)12-s + (0.286 − 0.957i)13-s + (−0.396 − 0.918i)14-s + (−0.597 − 0.802i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.396 + 0.918i)3-s + (0.173 − 0.984i)4-s + (−0.973 + 0.230i)5-s + (−0.893 − 0.448i)6-s + (−0.286 + 0.957i)7-s + (0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (0.597 − 0.802i)10-s + (0.597 + 0.802i)11-s + (0.973 − 0.230i)12-s + (0.286 − 0.957i)13-s + (−0.396 − 0.918i)14-s + (−0.597 − 0.802i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.429 - 0.903i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.429 - 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1394903302 + 0.08812613310i\)
\(L(\frac12)\) \(\approx\) \(-0.1394903302 + 0.08812613310i\)
\(L(1)\) \(\approx\) \(0.4454096113 + 0.4286718081i\)
\(L(1)\) \(\approx\) \(0.4454096113 + 0.4286718081i\)

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 \( 1 + (0.396 + 0.918i)T \)
5 \( 1 + (-0.973 + 0.230i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (0.597 + 0.802i)T \)
13 \( 1 + (0.286 - 0.957i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (-0.973 - 0.230i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (-0.0581 + 0.998i)T \)
53 \( 1 + (-0.286 + 0.957i)T \)
59 \( 1 + (-0.597 - 0.802i)T \)
61 \( 1 + (-0.893 - 0.448i)T \)
67 \( 1 + (-0.686 + 0.727i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.597 + 0.802i)T \)
79 \( 1 + (-0.973 - 0.230i)T \)
83 \( 1 + (0.597 - 0.802i)T \)
89 \( 1 + (0.0581 + 0.998i)T \)
97 \( 1 + (0.286 - 0.957i)T \)
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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

−18.17250942654185556283805203150, −17.20914925815442738501683689705, −16.597976715447125014218634944076, −16.3538179861650914124066689573, −15.18938442345004667362350481914, −14.346753297953880307176267543359, −13.571989228023736280404592222232, −13.08318514920772620549839127725, −12.27958362613874468688058796148, −11.64820417207093012014217716970, −11.19775759503385158927110465062, −10.44362077066017987662677250623, −9.23644237910337799259459170185, −8.990521772793471658214545840945, −8.07428780548086767008871380472, −7.64862402457652023447944494633, −6.725095884514478683681273619341, −6.54825081159790460683463003390, −4.879611241412542261228081264100, −3.921002316061739675100186250424, −3.410751401617605777861244213220, −2.77060413182123014319840917359, −1.44039030579767611599897986075, −1.05425119231757106778739250995, −0.06653216340182168602468243904, 1.39635376399673968219765052094, 2.49555629862343808873804866014, 3.258481367470238544878355276843, 4.09706918462936970666566553297, 4.960934387100053667334491223321, 5.64292509996866017786261395741, 6.43541943727658797671986258706, 7.34373361448522105625618117292, 8.070858610507056931293883367623, 8.59412155272961708798418095228, 9.182224245918526478566582751051, 9.985265498916379265083588737893, 10.57885402142086100084297656222, 11.162756121606189743530313939947, 12.175135641783413842747438431201, 12.659531888629468138710191352745, 14.03830270932043498342639495443, 14.73309235558031922886580996381, 15.08297405141509744841439477116, 15.5760328277488630551861019596, 16.17263079139900336207551488413, 16.91398467319231868062198104322, 17.47172286241732340805145206389, 18.576526659063810559719706149870, 18.92017850413091293991956108158

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