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Properties

Label 1-4029-4029.2957-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.515 + 0.857i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (0.200 − 0.979i)2-s + (−0.919 − 0.391i)4-s + (0.692 − 0.721i)5-s + (−0.996 + 0.0804i)7-s + (−0.568 + 0.822i)8-s + (−0.568 − 0.822i)10-s + (0.278 − 0.960i)11-s + (0.799 − 0.600i)13-s + (−0.120 + 0.992i)14-s + (0.692 + 0.721i)16-s + (−0.0402 + 0.999i)19-s + (−0.919 + 0.391i)20-s + (−0.885 − 0.464i)22-s + (−0.5 + 0.866i)23-s + (−0.0402 − 0.999i)25-s + (−0.428 − 0.903i)26-s + ⋯
L(s)  = 1  + (0.200 − 0.979i)2-s + (−0.919 − 0.391i)4-s + (0.692 − 0.721i)5-s + (−0.996 + 0.0804i)7-s + (−0.568 + 0.822i)8-s + (−0.568 − 0.822i)10-s + (0.278 − 0.960i)11-s + (0.799 − 0.600i)13-s + (−0.120 + 0.992i)14-s + (0.692 + 0.721i)16-s + (−0.0402 + 0.999i)19-s + (−0.919 + 0.391i)20-s + (−0.885 − 0.464i)22-s + (−0.5 + 0.866i)23-s + (−0.0402 − 0.999i)25-s + (−0.428 − 0.903i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.515 + 0.857i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (2957, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.515 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05631874031 + 0.03186113810i\)
\(L(\frac12)\) \(\approx\) \(0.05631874031 + 0.03186113810i\)
\(L(1)\) \(\approx\) \(0.6958617814 - 0.5528061117i\)
\(L(1)\) \(\approx\) \(0.6958617814 - 0.5528061117i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.200 - 0.979i)T \)
5 \( 1 + (0.692 - 0.721i)T \)
7 \( 1 + (-0.996 + 0.0804i)T \)
11 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (0.799 - 0.600i)T \)
19 \( 1 + (-0.0402 + 0.999i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.987 - 0.160i)T \)
31 \( 1 + (-0.948 + 0.316i)T \)
37 \( 1 + (-0.845 - 0.534i)T \)
41 \( 1 + (0.970 + 0.239i)T \)
43 \( 1 + (-0.278 - 0.960i)T \)
47 \( 1 + (-0.845 + 0.534i)T \)
53 \( 1 + (0.428 + 0.903i)T \)
59 \( 1 + (-0.919 + 0.391i)T \)
61 \( 1 + (0.885 - 0.464i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.568 + 0.822i)T \)
73 \( 1 + (-0.799 - 0.600i)T \)
83 \( 1 + (0.919 + 0.391i)T \)
89 \( 1 + (-0.568 - 0.822i)T \)
97 \( 1 + (-0.885 + 0.464i)T \)
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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

−18.064765711147559442102324378225, −17.80636534563793256040503331907, −16.78942255788272411022664384633, −16.42220456768586357805988722457, −15.59413215178975560134268112768, −14.92963058108253702711358477125, −14.41144594046801928360069083335, −13.59182585427285967477006968361, −13.13585908679549150736283293051, −12.515921442334131728405980080923, −11.5234765971571006549508094652, −10.60947041685406882114079631745, −9.817468314022522913126712663319, −9.31775694226131550181560338602, −8.7302581658571694613137998866, −7.58117742583902058350330896766, −6.90483479132277097117703479349, −6.515494080749373431438351984009, −5.89130866423643986849419879552, −5.0064759344664606685814996681, −4.09936072729962032561061053492, −3.459702179801343646553444160210, −2.55710292060284752288962357243, −1.5706309713931479982764788743, −0.01729078757279375521984716898, 1.10659897752040845630471413541, 1.73511155060698179935082742433, 2.751838748232918642044152282716, 3.61869574787592032429467147714, 3.93113341503430082162204561589, 5.24482399145595543498931312753, 5.85293394175220167732800584313, 6.10808561741277591697601273183, 7.53537481238589535038501697906, 8.54273722603031289949840453789, 8.98907546518782008343672385033, 9.65501441951918346531848019098, 10.30929324127135415444009218119, 10.96051010276795860974332556293, 11.807783892597269999533263332141, 12.56526441927213286514025559462, 13.00077594455238278468359551015, 13.66744573149920285107770639439, 14.09941021285101938926813059278, 15.087401392952351559614248397474, 16.086546405085666627015578310, 16.48479057822473706200303911344, 17.35911247829866411581494187582, 18.02576817954159373912428722335, 18.73817627911813211426440899423

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