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Properties

Label 1-4033-4033.289-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.130 + 0.991i$
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  + (−0.939 − 0.342i)2-s + (−0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (0.5 − 0.866i)5-s + (0.173 + 0.984i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + (0.173 − 0.984i)12-s + (−0.5 − 0.866i)13-s + (0.766 − 0.642i)14-s − 15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (0.5 − 0.866i)5-s + (0.173 + 0.984i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + (0.173 − 0.984i)12-s + (−0.5 − 0.866i)13-s + (0.766 − 0.642i)14-s − 15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.130 + 0.991i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.130 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2642064696 - 0.3014000185i\)
\(L(\frac12)\) \(\approx\) \(-0.2642064696 - 0.3014000185i\)
\(L(1)\) \(\approx\) \(0.3789609920 - 0.3506917420i\)
\(L(1)\) \(\approx\) \(0.3789609920 - 0.3506917420i\)

Euler product

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

−18.7763925774596463277945420223, −18.179735030741117596448763882543, −17.43847125421645994171259782419, −17.02266900579747216716728775106, −16.29975306265402099565073698693, −15.84291736404715437301708130399, −14.931992689344375060594217096, −14.45078364073233793147181996568, −13.87682377673490291926049594111, −12.61257615842687561890731670469, −11.89600609831545382065120232636, −10.997541296530456922311573254466, −10.406242721829015100639430645113, −10.07763600600180083650245580500, −9.55818317962419244165187912643, −8.75277737108810428431384808008, −7.4707285636608284154598477276, −7.27654880693173906450718204507, −6.34952157909298774374025504688, −5.71441581460777360842657402236, −5.04599251086301209897939665337, −3.83828509177425564551797622014, −3.26359720159801454576314804342, −2.152402143900848373430388488800, −1.33585564488073503275903888381, 0.23063274107112838691504493242, 0.720893903610120624956858259158, 1.855522698166402553660835246143, 2.60612618159996973609439702329, 3.02554896504435280275674999910, 4.61577522677774573400478058713, 5.6574566853562038135986351834, 5.805383267097238452880746033162, 6.829272753903592423665784892771, 7.68297823256819810541376596277, 8.25857484933306713399742928101, 8.82763798544806397114787910431, 9.76671930773445894271965413602, 10.158407312759257856948252138181, 11.25208845060407052937915654702, 11.76005479125759355359997399006, 12.45761551449332648390944290325, 13.072857902423895594408967823, 13.34024736761425646774687281797, 14.61896370514483716555373867983, 15.621348292633395388288815494036, 16.28862486137459943077729434697, 16.64391214173211679981013372370, 17.482177598764443334291033030236, 18.10207147456271655221923111924

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