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Properties

Label 1-4033-4033.256-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.826 + 0.563i$
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 + 5-s + (0.173 + 0.984i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + (0.766 + 0.642i)12-s + (−0.5 + 0.866i)13-s + (0.766 − 0.642i)14-s + (−0.5 + 0.866i)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 + 5-s + (0.173 + 0.984i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + (0.766 + 0.642i)12-s + (−0.5 + 0.866i)13-s + (0.766 − 0.642i)14-s + (−0.5 + 0.866i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.826 + 0.563i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.826 + 0.563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.854462802 + 0.8805900388i\)
\(L(\frac12)\) \(\approx\) \(2.854462802 + 0.8805900388i\)
\(L(1)\) \(\approx\) \(1.737565423 + 0.01876444844i\)
\(L(1)\) \(\approx\) \(1.737565423 + 0.01876444844i\)

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

−18.0927082012908361662771572811, −17.55061168898715597351408804534, −17.06883470201148177922421280841, −16.69759654232514237043693966654, −15.54587728445229865846396461330, −14.87300050953651497735406674552, −14.091849599848676013984715562437, −13.64848855296549176691957671374, −13.23620912767854588556732881646, −12.21972715593221811031701221025, −11.78547880225541847359542921057, −11.12115265622168471944360872396, −10.272029333784441537141059140469, −9.16385418188064112291092553612, −8.39421756689947299784072104353, −7.73365423906033043417246324840, −7.1106401559081932026445356495, −6.12312100967626064949589916289, −5.94220716911089202629788851779, −5.01731501896456893833086746516, −4.578053573645087134357798333869, −3.256118200637817415823805104700, −2.410866757381001318271718085216, −1.78255629296073700343524006246, −0.655016030871816460255165609110, 1.21977150276892814873161267092, 1.831767664713539585485317216188, 2.56302798937848806813444353398, 3.7795962587959666143636430054, 4.34587097400758448405982280754, 4.929229708296107265253214733272, 5.54939934635365652890945782669, 6.507533705598667808473419220926, 6.73969481365588569957620805775, 8.417306703729789191483230816930, 9.055573986235987843286453936647, 9.81163375738649709953886955883, 10.49626412624234591090632272942, 10.74719557281664560530538303794, 11.8834283906856171528772268515, 12.122207872102907573692794555198, 12.93342249632389937192156739045, 14.039937439769316472240359735054, 14.48337703232047320660204617820, 14.77514180933734449883928225037, 15.60230337788931150839787573361, 16.73084962719832852954683351409, 17.04974768933759001065124845449, 17.790051868848776567600828247194, 18.48076726820875176822142392533

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