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Properties

Label 1-4033-4033.978-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.997 + 0.0688i$
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 + (0.893 − 0.448i)3-s + (0.766 + 0.642i)4-s + (0.597 + 0.802i)5-s + (0.993 − 0.116i)6-s + (0.396 − 0.918i)7-s + (0.5 + 0.866i)8-s + (0.597 − 0.802i)9-s + (0.286 + 0.957i)10-s + (0.286 − 0.957i)11-s + (0.973 + 0.230i)12-s + (0.993 − 0.116i)13-s + (0.686 − 0.727i)14-s + (0.893 + 0.448i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.893 − 0.448i)3-s + (0.766 + 0.642i)4-s + (0.597 + 0.802i)5-s + (0.993 − 0.116i)6-s + (0.396 − 0.918i)7-s + (0.5 + 0.866i)8-s + (0.597 − 0.802i)9-s + (0.286 + 0.957i)10-s + (0.286 − 0.957i)11-s + (0.973 + 0.230i)12-s + (0.993 − 0.116i)13-s + (0.686 − 0.727i)14-s + (0.893 + 0.448i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.997 + 0.0688i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (978, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.997 + 0.0688i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.363145097 + 0.2191598292i\)
\(L(\frac12)\) \(\approx\) \(6.363145097 + 0.2191598292i\)
\(L(1)\) \(\approx\) \(3.111657483 + 0.2155309797i\)
\(L(1)\) \(\approx\) \(3.111657483 + 0.2155309797i\)

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 \( 1 + (0.893 - 0.448i)T \)
5 \( 1 + (0.597 + 0.802i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (0.286 - 0.957i)T \)
13 \( 1 + (0.993 - 0.116i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 - T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.286 - 0.957i)T \)
31 \( 1 + (-0.0581 - 0.998i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.597 + 0.802i)T \)
53 \( 1 + (-0.973 - 0.230i)T \)
59 \( 1 + (-0.893 - 0.448i)T \)
61 \( 1 + (0.973 + 0.230i)T \)
67 \( 1 + (0.286 - 0.957i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.973 - 0.230i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.893 - 0.448i)T \)
97 \( 1 + (-0.835 - 0.549i)T \)
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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

−18.69890679531307452902434847387, −17.90625601324722150844848337059, −16.80304841453178913517585542379, −16.26699774992952694054943487081, −15.47993055790091384234912046009, −14.93541921984690296915428820877, −14.382854143255386792013004264065, −13.74016911828410686803417732484, −12.93125656110268060306033716338, −12.5418048256104248227154673696, −11.82397417230950771899536317630, −10.83695694756984278857476746674, −10.20686741525898458641052068701, −9.43856758324191742521082196354, −8.812837029214561470980507911870, −8.252004426259010133541700132919, −7.11057981506539111010392489657, −6.37884976121186179539045398532, −5.25976504039191444689167882863, −5.06613609432420958131997293869, −4.226036552076472820822309873181, −3.40056699183620596743705529822, −2.63119674512385846250114670921, −1.68748890975599870905191120048, −1.48389800519987949926171516615, 1.18839191953958273357098701568, 1.8345257976879875165893546322, 2.83568517973520353419606597279, 3.51041381819709607955542102659, 3.87862525989799540285312994401, 4.97695410100969144738667218327, 6.18624598452165331322423416393, 6.28993668559345798841845306537, 7.18869303873186762596973836163, 7.97873834437023230865338172385, 8.330397188122096803465927118028, 9.45612435181595317605668065632, 10.31120050963952442145470839559, 11.13509712637342560166292376322, 11.47855117148281928961895906026, 12.77756885369140855850212334011, 13.26100760555663683517445937656, 13.75860913577797242087010379814, 14.26541796364133874548888373631, 14.90783564528440545282282552925, 15.34700550116867103961964547082, 16.46850212209437865682586710673, 17.04077235873644053024926978989, 17.66891819201517282040408650, 18.68837986159069419797179181169

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