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Properties

Label 1-4033-4033.169-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.741 - 0.670i$
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 + (−0.835 + 0.549i)3-s + (−0.5 − 0.866i)4-s + (−0.597 + 0.802i)5-s + (−0.0581 − 0.998i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (−0.396 − 0.918i)10-s + (−0.396 + 0.918i)11-s + (0.893 + 0.448i)12-s + (0.597 − 0.802i)13-s + (−0.993 − 0.116i)14-s + (0.0581 − 0.998i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.835 + 0.549i)3-s + (−0.5 − 0.866i)4-s + (−0.597 + 0.802i)5-s + (−0.0581 − 0.998i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (−0.396 − 0.918i)10-s + (−0.396 + 0.918i)11-s + (0.893 + 0.448i)12-s + (0.597 − 0.802i)13-s + (−0.993 − 0.116i)14-s + (0.0581 − 0.998i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.741 - 0.670i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.741 - 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1852610322 + 0.4813801863i\)
\(L(\frac12)\) \(\approx\) \(-0.1852610322 + 0.4813801863i\)
\(L(1)\) \(\approx\) \(0.3808306645 + 0.3967821512i\)
\(L(1)\) \(\approx\) \(0.3808306645 + 0.3967821512i\)

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 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (-0.597 + 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (-0.396 + 0.918i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (0.686 - 0.727i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.0581 - 0.998i)T \)
53 \( 1 + (0.0581 + 0.998i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (0.686 + 0.727i)T \)
67 \( 1 + (-0.893 + 0.448i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.993 + 0.116i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (-0.396 - 0.918i)T \)
97 \( 1 + (0.686 + 0.727i)T \)
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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

−17.994476247002783693063347305472, −17.34945420462840902349294995398, −16.94664563230604638720442746818, −16.23619209872502454369183448535, −15.81030889681005936736497746075, −14.34293007989848655567967137454, −13.63148563623692070086676996471, −12.89903997598591603581426451725, −12.68815528561435149280711764172, −11.62783697823462206682198885038, −11.141392436208900274629881108774, −10.79427160127967949113282006892, −9.99967075597632100670644307856, −8.72867756335849911775319646326, −8.45325650259471597108833846788, −7.77215505095651660691586550926, −6.87233015721785767923223751214, −6.2072339633389370783824104224, −4.92271808164986135066091659235, −4.48088001990692136388169804561, −3.85279678391639187262458209796, −2.72769635282784557307691795054, −1.61766462888511095752401882558, −1.06591781472607632660829961484, −0.28379995228009192733042485828, 0.905661776405346678816829541681, 2.17318028543063374120774201980, 3.12999245332905682120627079466, 4.37618528064503648818265773197, 4.70532768441923904231692168003, 5.7367165805257760005324608169, 6.09456038201078709996761485387, 7.078343015938997162979180331381, 7.511779969488778095679256135162, 8.53358411470151290679047409140, 9.02777388518227767224431840389, 10.17379948530993458510093013806, 10.34201857785017573156928790336, 11.27060516225564591063233462619, 11.77632134171289945447725207750, 12.6773626253778948238971765712, 13.52550901625408432184659886188, 14.65817757315963615589237089239, 15.06062322899111875526550173573, 15.61220647382511483697181835859, 15.86984416611139866477023864460, 16.89349100788566421074477582810, 17.71155266488425231966059062329, 17.976682577613862195959871552345, 18.5841482578243418127703197881

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