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Properties

Label 1-4033-4033.468-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.399 - 0.916i$
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 + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.866 + 0.5i)5-s + (−0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (−0.766 − 0.642i)12-s + (−0.5 − 0.866i)13-s + (−0.939 − 0.342i)14-s + i·15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.866 + 0.5i)5-s + (−0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (−0.766 − 0.642i)12-s + (−0.5 − 0.866i)13-s + (−0.939 − 0.342i)14-s + i·15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (468, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.399 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.201797488 + 1.835264155i\)
\(L(\frac12)\) \(\approx\) \(-1.201797488 + 1.835264155i\)
\(L(1)\) \(\approx\) \(0.5224647585 + 1.263253291i\)
\(L(1)\) \(\approx\) \(0.5224647585 + 1.263253291i\)

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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.342 + 0.939i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.642 - 0.766i)T \)
31 \( 1 + (0.342 + 0.939i)T \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (-0.984 - 0.173i)T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (-0.642 - 0.766i)T \)
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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

−18.28497635419137732776396499429, −17.35932379810735542786466104226, −16.88791883120586303164189239931, −16.258040077339127622410999716696, −14.79469626381313861572465593397, −14.168662677873432217148066320926, −13.781538510611355467582839964009, −13.26219181899544398711631992606, −12.62139079662982840606183064256, −11.94964097299602749172113629846, −11.25742250978221740526006794442, −10.37176309903003677494607693151, −9.56266873682250095169517537834, −9.17490557824302220302581259765, −8.50835553582425918839893747595, −7.5092168956122228020322280049, −6.73844078655530567463333556290, −5.953400616021632709615023821393, −5.20001071631421679782513995339, −4.25953394674629224746323217417, −3.39564842044323775617621340183, −2.781886827426288945993854424546, −1.9542111626870687550386063074, −1.078124126625894864363243779242, −0.59052500562118616385907973905, 1.49023316155127231843778915927, 2.65429163070816985479080416526, 3.208815904326198907366157324579, 3.948102240300652018852512847560, 5.15022654717960607819389858679, 5.40754878567536305404730921287, 6.1022692371056874543017333552, 7.05963664978411861954773546516, 7.75221485506351123312240752177, 8.493003763738392861567977771906, 9.44036829020558988200717921383, 9.76825607935126591024793592309, 10.117674604444591783506768595651, 11.34770079456904315701776308368, 12.42790040664121210083525119473, 12.86182639653228288861364052961, 13.80444624016815472607358004813, 14.41727668927221766815468069893, 14.85812077679078336414958043615, 15.51048460154471408896597696597, 15.94022636868659483158184931108, 17.02089102141492817607424997721, 17.28004634503356495104027330503, 18.09736310771417188815355259317, 18.85866358288110188153873809565

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