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Properties

Label 1-4033-4033.620-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0780 - 0.996i$
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 + (0.173 − 0.984i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + 12-s + (0.939 − 0.342i)13-s + (−0.766 + 0.642i)14-s + (0.939 + 0.342i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + 12-s + (0.939 − 0.342i)13-s + (−0.766 + 0.642i)14-s + (0.939 + 0.342i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0780 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0780 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0780 - 0.996i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (620, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0780 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9247304855 - 0.8551655282i\)
\(L(\frac12)\) \(\approx\) \(0.9247304855 - 0.8551655282i\)
\(L(1)\) \(\approx\) \(0.7395481971 - 0.3894542149i\)
\(L(1)\) \(\approx\) \(0.7395481971 - 0.3894542149i\)

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 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.766 + 0.642i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 - T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−18.725502711522571903083251188454, −17.675141569620100183715512783319, −17.2591134332091402430295205267, −16.22784116537663265226294012190, −16.01258860415568518512440395430, −15.72725516185187781980022306210, −14.593593897989267615025025586285, −14.17525890934211059912820933666, −13.47605937199787814191662879195, −12.14280778052195685147534834106, −11.64310830466615167053952970793, −11.03425487864510714090392842355, −10.01062092261608193228198768929, −9.437843994691911789585695721273, −8.87230228200321889397803239929, −8.34069310712733221955573820687, −7.94233022973601866761896653564, −6.45476969858707775084111293759, −5.95559661835254604963273795116, −5.29171768609992190965118563542, −4.52412866397474933559320196434, −3.82864948572485631472344574479, −2.6736232803336366151893133360, −1.70924774779910708158598445593, −0.755581922939932835388924860370, 0.65933146517347850610854767337, 1.435740841949901449957623175467, 2.20963846322285888565271386136, 2.991701006550528651634716097936, 3.81981241523868929643777089376, 4.343403666553781274082320877711, 6.080665518298640953955734234774, 6.63130800443280601309402867985, 7.16031357242924561584243724846, 7.9178107281512189762913264472, 8.38623657020187176103141523932, 9.30138172623578503728532272117, 10.18491485103254416051334792250, 10.8000598195903921624501489794, 11.28771501119968886836317184841, 12.07372002527888166714123684490, 12.670062677659102656262870779290, 13.66362649934849556252616927596, 13.83771058910012584214379469293, 14.88150020749396296121858936083, 15.56434642883666527642997693113, 16.58946904443793529295082491106, 17.34214210029377439397508568088, 17.93442890062755147572213945749, 18.0351402338697760943712853681

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