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Properties

Label 1-4033-4033.853-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.999 + 0.0291i$
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  + (0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.866 + 0.5i)5-s + (0.939 − 0.342i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.984 − 0.173i)10-s + (0.984 − 0.173i)11-s + (0.939 + 0.342i)12-s + (−0.5 + 0.866i)13-s + (0.173 − 0.984i)14-s + i·15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.866 + 0.5i)5-s + (0.939 − 0.342i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.984 − 0.173i)10-s + (0.984 − 0.173i)11-s + (0.939 + 0.342i)12-s + (−0.5 + 0.866i)13-s + (0.173 − 0.984i)14-s + i·15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.999 + 0.0291i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.999 + 0.0291i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.236105170 + 0.03259508429i\)
\(L(\frac12)\) \(\approx\) \(2.236105170 + 0.03259508429i\)
\(L(1)\) \(\approx\) \(1.451301344 + 0.1946631622i\)
\(L(1)\) \(\approx\) \(1.451301344 + 0.1946631622i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.984 - 0.173i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.342 + 0.939i)T \)
31 \( 1 + (0.984 - 0.173i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + (-0.642 - 0.766i)T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (0.342 - 0.939i)T \)
97 \( 1 + (0.342 + 0.939i)T \)
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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

−19.05573903542147722797020403338, −17.81933198924114310205029920030, −16.88593924540373075251471668297, −16.09578987667764655243505257744, −15.54686815817989224435579975270, −15.02005497782387153310663993126, −14.623491006359116791599564013288, −13.56881004128878113907655627695, −13.02791792521180948012755891153, −12.09945179545218309481047262337, −11.79541174702617979623539276251, −11.052484449803445030165362300389, −10.11969748706202431474160050513, −9.53262357848256584584211308617, −8.97288350270671448698046838919, −8.22165701568035150645419793697, −7.26156480253397326850746148670, −6.22819466785284516085749331872, −5.485479131103101526855840069850, −4.65140830010202321284870239478, −4.29959259453774365715463436523, −3.28546991478395055920824722413, −2.92470668848025196779335680615, −2.00341838436672784154718135479, −0.76553493499724758552369774182, 0.597312370544981129208542047800, 1.932043702049358630039457263280, 2.757194762404652565564015763750, 3.6737555162288590301703325470, 3.98142126259382861621353327926, 4.810325837889779298403364849943, 6.27888596742306289713579328789, 6.66670278027029903283038667506, 6.970681519957635496235279558180, 7.79932918676733218967729861650, 8.56102934990408663011015992924, 9.02569187984179159873050542921, 10.329999422635531606634057935650, 11.17157597732338946145215465297, 11.8554126355642137058004116630, 12.51366803155485544872185591071, 12.99405230159427857515144100055, 13.9140340131162635779981927357, 14.36290418276079434313252447505, 14.83713394496164434237655134576, 15.57492249052596048440629844374, 16.4458468575425214163006549393, 17.13675251144737600626759486299, 17.44625058313499655220111916525, 18.69901509625004945264842540555

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