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Properties

Label 1-4033-4033.4008-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.125 + 0.992i$
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 + (−0.835 + 0.549i)3-s + 4-s + (0.597 − 0.802i)5-s + (0.835 − 0.549i)6-s + (0.396 + 0.918i)7-s − 8-s + (0.396 − 0.918i)9-s + (−0.597 + 0.802i)10-s + (−0.597 − 0.802i)11-s + (−0.835 + 0.549i)12-s + (−0.597 + 0.802i)13-s + (−0.396 − 0.918i)14-s + (−0.0581 + 0.998i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.835 + 0.549i)3-s + 4-s + (0.597 − 0.802i)5-s + (0.835 − 0.549i)6-s + (0.396 + 0.918i)7-s − 8-s + (0.396 − 0.918i)9-s + (−0.597 + 0.802i)10-s + (−0.597 − 0.802i)11-s + (−0.835 + 0.549i)12-s + (−0.597 + 0.802i)13-s + (−0.396 − 0.918i)14-s + (−0.0581 + 0.998i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.125 + 0.992i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (4008, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.125 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4462107094 + 0.3932582131i\)
\(L(\frac12)\) \(\approx\) \(0.4462107094 + 0.3932582131i\)
\(L(1)\) \(\approx\) \(0.5523232816 + 0.07701866020i\)
\(L(1)\) \(\approx\) \(0.5523232816 + 0.07701866020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (-0.597 - 0.802i)T \)
13 \( 1 + (-0.597 + 0.802i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.396 + 0.918i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.835 + 0.549i)T \)
53 \( 1 + (-0.893 - 0.448i)T \)
59 \( 1 + (0.0581 - 0.998i)T \)
61 \( 1 + (-0.686 - 0.727i)T \)
67 \( 1 + (0.835 + 0.549i)T \)
71 \( 1 + T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (0.0581 - 0.998i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (-0.993 - 0.116i)T \)
97 \( 1 + (-0.286 + 0.957i)T \)
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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

−18.081435589057788853833968553419, −17.60823592653533422599426074394, −17.25888042385889657964857218556, −16.73583137561062832142707787817, −15.46660603721344349593765803931, −15.31359283311048235370422550855, −14.16516511292093172269371575773, −13.5351509156832535306487745834, −12.6230719689050022505226926467, −12.07421547186873216920784733188, −11.13563937659894049263138759174, −10.56872064931672925993872066484, −10.23096206285885363306161062174, −9.63758894952838155151193469945, −8.32117593693274799719438773099, −7.66800820892975426535417052713, −7.15009869827038376868128792734, −6.667898733001117043532113439465, −5.701279024345718229107045479409, −5.188643218436665521693995488668, −3.97780164381037046395124190195, −2.8129644090233240615281995987, −2.07651615418979456674007400838, −1.424609615799820887393360834060, −0.34127456772388463650731742592, 0.77592495959904368724998109294, 1.71716939674136356011354744439, 2.46189924722086205319480295908, 3.48676275407976238850131887492, 4.76042258335640977203063519315, 5.226374936707063950713886546563, 6.06995767853960128516516556936, 6.40137004991612552337917265459, 7.69982972126838688324954095992, 8.299824302424363280919900425791, 9.11603328342773052274718074246, 9.58354245923755730282615124249, 10.17826044307254677303638077853, 11.02757373124543949771212453324, 11.655392518196631302357208550688, 12.27384565417002461492408046627, 12.72462070869111064937068329640, 14.09625604180381455851838356718, 14.64644373988349909841043591368, 15.63391316254264838328731304334, 16.26945082825703098227524543554, 16.50712854719333051581398494034, 17.21015231024219181615800999430, 17.94552865042032041881046818892, 18.55139339660159913894496498254

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