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Properties

Label 1-6017-6017.2604-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.545 - 0.838i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (0.498 − 0.867i)2-s + (0.963 + 0.266i)3-s + (−0.503 − 0.863i)4-s + (−0.804 − 0.593i)5-s + (0.710 − 0.703i)6-s + (0.999 − 0.0276i)7-s + (−0.999 + 0.00690i)8-s + (0.858 + 0.512i)9-s + (−0.915 + 0.402i)10-s + (−0.256 − 0.966i)12-s + (−0.134 − 0.990i)13-s + (0.473 − 0.880i)14-s + (−0.618 − 0.786i)15-s + (−0.492 + 0.870i)16-s + (0.436 + 0.899i)17-s + (0.872 − 0.488i)18-s + ⋯
L(s)  = 1  + (0.498 − 0.867i)2-s + (0.963 + 0.266i)3-s + (−0.503 − 0.863i)4-s + (−0.804 − 0.593i)5-s + (0.710 − 0.703i)6-s + (0.999 − 0.0276i)7-s + (−0.999 + 0.00690i)8-s + (0.858 + 0.512i)9-s + (−0.915 + 0.402i)10-s + (−0.256 − 0.966i)12-s + (−0.134 − 0.990i)13-s + (0.473 − 0.880i)14-s + (−0.618 − 0.786i)15-s + (−0.492 + 0.870i)16-s + (0.436 + 0.899i)17-s + (0.872 − 0.488i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (2604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.545 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.725294524 - 1.478255725i\)
\(L(\frac12)\) \(\approx\) \(2.725294524 - 1.478255725i\)
\(L(1)\) \(\approx\) \(1.545693598 - 0.7588793727i\)
\(L(1)\) \(\approx\) \(1.545693598 - 0.7588793727i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.498 - 0.867i)T \)
3 \( 1 + (0.963 + 0.266i)T \)
5 \( 1 + (-0.804 - 0.593i)T \)
7 \( 1 + (0.999 - 0.0276i)T \)
13 \( 1 + (-0.134 - 0.990i)T \)
17 \( 1 + (0.436 + 0.899i)T \)
19 \( 1 + (-0.993 + 0.110i)T \)
23 \( 1 + (0.539 + 0.842i)T \)
29 \( 1 + (-0.865 - 0.500i)T \)
31 \( 1 + (-0.242 + 0.970i)T \)
37 \( 1 + (0.562 + 0.826i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.188 - 0.982i)T \)
47 \( 1 + (0.485 + 0.873i)T \)
53 \( 1 + (0.556 - 0.830i)T \)
59 \( 1 + (0.943 - 0.331i)T \)
61 \( 1 + (0.945 - 0.325i)T \)
67 \( 1 + (0.256 + 0.966i)T \)
71 \( 1 + (0.639 + 0.768i)T \)
73 \( 1 + (0.938 + 0.344i)T \)
79 \( 1 + (-0.875 - 0.482i)T \)
83 \( 1 + (0.399 - 0.916i)T \)
89 \( 1 + (0.0172 - 0.999i)T \)
97 \( 1 + (0.878 - 0.476i)T \)
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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

−17.99708961063000505699268170411, −16.885914289023794028749286879927, −16.4904484433336827835533229089, −15.58207040153103915161984188457, −14.92268835961330373975895125537, −14.69642180530077264319654233869, −14.110485753432236285133163143209, −13.49917762770150071084500538050, −12.653349292377079462004991509162, −12.04060260258793470868212602338, −11.38565972565452539761348947757, −10.64778417752468467637584366748, −9.46116606817736958053549507901, −8.9282027811574763592290147037, −8.208679017757063657814957037953, −7.75261433163007013703131654772, −6.98466695010034226168782057222, −6.77001986133763963972338458127, −5.601114185186011276636521575548, −4.67654162255842607544704464113, −4.13823334828477807987832468402, −3.60547142649323057012601962648, −2.57216870255987235287447825736, −2.12788231826148797146244172352, −0.68787204363900846076992400435, 0.89331689908771691183449862232, 1.6067701559373826222969173529, 2.309584082616864159258713578730, 3.313305533316416658775183843283, 3.769016996585787143215043799771, 4.43914600919330574671523619311, 5.1087608832349515731648800819, 5.67246855531512656446980411131, 7.00208884841754726257414131444, 7.842026321719796270411643783831, 8.43476632473736554302699647180, 8.76327975999585250841908203164, 9.79027122045270579709596329883, 10.34572584336440021028152605273, 11.08145307689917526098159582316, 11.61079223431770784192059768564, 12.56815089965420921221613797494, 12.89888605385174923434651626592, 13.54559398337222515266983685599, 14.44633771971249748273282730145, 14.98791041003339226208320937641, 15.21562053078294069971199835704, 16.0291816032928913377292243270, 17.11356891734344654899795947320, 17.58795698698995954124811247409

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