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Properties

Label 1-4033-4033.2293-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.999 + 0.0231i$
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  + (−0.5 − 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.5 + 0.866i)4-s + (−0.173 − 0.984i)5-s + (−0.939 − 0.342i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + (0.173 + 0.984i)12-s + (0.173 + 0.984i)13-s + (0.766 − 0.642i)14-s + (−0.766 − 0.642i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.5 + 0.866i)4-s + (−0.173 − 0.984i)5-s + (−0.939 − 0.342i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + (0.173 + 0.984i)12-s + (0.173 + 0.984i)13-s + (0.766 − 0.642i)14-s + (−0.766 − 0.642i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.999 + 0.0231i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.999 + 0.0231i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7560638919 + 0.008746717246i\)
\(L(\frac12)\) \(\approx\) \(0.7560638919 + 0.008746717246i\)
\(L(1)\) \(\approx\) \(0.6932883545 - 0.4470532272i\)
\(L(1)\) \(\approx\) \(0.6932883545 - 0.4470532272i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
41 \( 1 - T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−18.557407391858507147959629431167, −17.67080770328444889642173543122, −17.10863464874166286892139731640, −16.383020226423670374766375511944, −15.52639757727486921965510807451, −15.10316745213145716217278221084, −14.71862954297684162344835625891, −13.87536308276345294655370729114, −13.40530382402293799588430408218, −12.56008676794845014107921092023, −10.99202424428206887151840570866, −10.60676530474715348376125867634, −10.20140797383338467968189696216, −9.586029734260650029761547739382, −8.48307440928621530635371177047, −7.79848463833131541683242859740, −7.67006283274733306168772274181, −6.66419869678836829683642223695, −5.90738287420085306381377678928, −5.00402864635463797551438218854, −4.02461176584359783808915881125, −3.78052557092515519355622197779, −2.44569779216862060459099840912, −1.83412697102601528592288852261, −0.25139572354696293464657426547, 0.91169611375177470845569076667, 1.81263390317127741571231081874, 2.33916022609400470506222831169, 3.15003589348622898690519660401, 3.97557143331467312222667370459, 4.86785269302594218796027641604, 5.58548647500869386238824423661, 6.81287158709914475846920286803, 7.526562425207292518484067667315, 8.39839203585654130101129619025, 8.78720282691240294020042439811, 9.174755516409895604515712248765, 9.93796101386797693313109243867, 11.21004739020308193369099785087, 11.61370171439950567196621338229, 12.26614242623983046263603638241, 13.01472132522446657896134383504, 13.404564177816928574632803142209, 14.071084041714206899141458826441, 15.07505324092689010239479935537, 16.03458182275469209304909536699, 16.25581798994998498481339104745, 17.53433328584716355753559866571, 17.88210069479373605164869731232, 18.695511627297850104510866923950

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