Properties

Label 1-4033-4033.1743-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.252 - 0.967i$
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 + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + 6-s + (0.173 − 0.984i)7-s + (−0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.939 − 0.342i)12-s + (0.766 + 0.642i)13-s + (−0.5 + 0.866i)14-s + (−0.173 − 0.984i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + 6-s + (0.173 − 0.984i)7-s + (−0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.939 − 0.342i)12-s + (0.766 + 0.642i)13-s + (−0.5 + 0.866i)14-s + (−0.173 − 0.984i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1438604945 - 0.1862857917i\)
\(L(\frac12)\) \(\approx\) \(0.1438604945 - 0.1862857917i\)
\(L(1)\) \(\approx\) \(0.4871929390 + 0.04336686099i\)
\(L(1)\) \(\approx\) \(0.4871929390 + 0.04336686099i\)

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

−18.57463126060815821043541296489, −17.97405061655235152629132893230, −17.20510640638450414281776822356, −16.71811190450095657911060433129, −16.21906282115995598063654451504, −15.51432905982359382621148817721, −14.94953855934480645019410856922, −13.86950153497771445990838100697, −12.9670581177521465930631777361, −12.27603893284721990072131316331, −11.81785892451478179577870566439, −11.01534388387599576751988577050, −10.511520887011174760089368351420, −9.53425525795776520193537664856, −8.80390140817164558056678270318, −8.17244905634577956466190860170, −7.845769833723611770795204171556, −6.38642173658875692898521092263, −6.19787878559350779745475804265, −5.48530967509513160102872684134, −4.80396157877554671496939237274, −3.69071948011102627703850629531, −2.41662799319704690744607560560, −1.52036097360456066844627226438, −0.91442093342074426886532531586, 0.1274348472112161869709147660, 1.414190127718094416167035878225, 1.89508637421088011540656701710, 3.40438147172711758190550276992, 3.742851267005914978009599172413, 4.56900339714972619180133774086, 5.77240990402238799278963353728, 6.6354407466559500774291172394, 7.049717491191229985318124449721, 7.56428308221236999337622960944, 8.64278824253554831153333122510, 9.54536516658558586796464105246, 10.24672237115909383653520303756, 10.48036628611663060750113664839, 11.373803739943390728282129953360, 11.70625900176739707075916121779, 12.44754137338122480914151073651, 13.40870878786759916423018789773, 14.3453205177381450428091415521, 15.024666828269300884523532944957, 15.78949033098621178917104658139, 16.52872164296138158214371004985, 16.904016990751420989666872266503, 17.75065380289631899371450212194, 18.11894584229844306026819746903

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