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Properties

Label 1-4033-4033.24-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.0326 - 0.999i$
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  − 2-s + (−0.396 + 0.918i)3-s + 4-s + (−0.957 + 0.286i)5-s + (0.396 − 0.918i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (0.957 − 0.286i)10-s + (−0.957 − 0.286i)11-s + (−0.396 + 0.918i)12-s + (0.286 + 0.957i)13-s + (0.686 − 0.727i)14-s + (0.116 − 0.993i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.396 + 0.918i)3-s + 4-s + (−0.957 + 0.286i)5-s + (0.396 − 0.918i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (0.957 − 0.286i)10-s + (−0.957 − 0.286i)11-s + (−0.396 + 0.918i)12-s + (0.286 + 0.957i)13-s + (0.686 − 0.727i)14-s + (0.116 − 0.993i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0326 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0326 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.0326 - 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.0326 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1243635628 + 0.1284907814i\)
\(L(\frac12)\) \(\approx\) \(-0.1243635628 + 0.1284907814i\)
\(L(1)\) \(\approx\) \(0.3362706049 + 0.2192599153i\)
\(L(1)\) \(\approx\) \(0.3362706049 + 0.2192599153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.396 + 0.918i)T \)
5 \( 1 + (-0.957 + 0.286i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (-0.957 - 0.286i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.727 + 0.686i)T \)
31 \( 1 + (0.549 + 0.835i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.918 + 0.396i)T \)
53 \( 1 + (-0.802 + 0.597i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (0.998 + 0.0581i)T \)
67 \( 1 + (-0.918 + 0.396i)T \)
71 \( 1 - T \)
73 \( 1 + (0.686 + 0.727i)T \)
79 \( 1 + (0.993 + 0.116i)T \)
83 \( 1 + (-0.893 + 0.448i)T \)
89 \( 1 + (-0.230 + 0.973i)T \)
97 \( 1 + (-0.549 + 0.835i)T \)
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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

−17.89170573997179846342738251674, −17.50132818055455343688247234448, −16.58639219013946885172233360195, −16.1974752669330369639417156085, −15.461737557169137226179241017849, −14.86381783443558403771752365340, −13.55873656193185691866900025701, −12.95914313973377136401427396945, −12.50817893601779414540070159698, −11.63876291483353613382078065536, −11.07225333239434621633638340531, −10.37133318251465147473027390150, −9.82110304036272708442265004540, −8.54925432758103738308029133678, −8.110750406212808674822973346919, −7.63707772980790839173563287441, −6.843551299720423214420068864294, −6.341772051723443977092319824876, −5.38815091496642917664621880140, −4.45072757697225147889943599296, −3.22178337198068639150925829975, −2.749851126512220705624973526599, −1.58782526444239784444723288487, −0.64173392877324068024552203705, −0.13326033663394348675450062582, 1.12297723603353332018706479637, 2.55453496232978239372206293580, 3.13090971945304063001429768184, 3.73491894415442639266014880583, 4.97846758020171758364886748552, 5.51976562454552221880215225504, 6.61293631156581745921545049534, 6.97077523365755519305625372578, 8.0466415678153582134623511196, 8.69394837761773369128582829028, 9.29011610059175085188455840047, 9.97731248680740227412775052934, 10.633199522090498811410057576787, 11.40499018246386773130018726682, 11.79071588058903084876769173436, 12.361221396234228513649887192143, 13.59083061390302821811331902056, 14.52551875401638966388648569624, 15.379355544461449199768459122450, 15.90656354904503796000866462589, 16.02730414355467531163615574313, 16.69255597357994912012310238301, 17.824989022848339032694584426303, 18.21513953599116885423273933602, 18.98254599307310691957595693959

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