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Properties

Label 1-6041-6041.419-r0-0-0
Degree $1$
Conductor $6041$
Sign $0.0992 + 0.995i$
Analytic cond. $28.0542$
Root an. cond. $28.0542$
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  + (0.325 − 0.945i)2-s + (−0.999 + 0.0291i)3-s + (−0.787 − 0.615i)4-s + (−0.400 + 0.916i)5-s + (−0.297 + 0.954i)6-s + (−0.838 + 0.544i)8-s + (0.998 − 0.0582i)9-s + (0.736 + 0.676i)10-s + (0.177 − 0.984i)11-s + (0.805 + 0.592i)12-s + (0.0691 + 0.997i)13-s + (0.373 − 0.927i)15-s + (0.241 + 0.970i)16-s + (−0.339 + 0.940i)17-s + (0.269 − 0.962i)18-s + (−0.801 − 0.598i)19-s + ⋯
L(s)  = 1  + (0.325 − 0.945i)2-s + (−0.999 + 0.0291i)3-s + (−0.787 − 0.615i)4-s + (−0.400 + 0.916i)5-s + (−0.297 + 0.954i)6-s + (−0.838 + 0.544i)8-s + (0.998 − 0.0582i)9-s + (0.736 + 0.676i)10-s + (0.177 − 0.984i)11-s + (0.805 + 0.592i)12-s + (0.0691 + 0.997i)13-s + (0.373 − 0.927i)15-s + (0.241 + 0.970i)16-s + (−0.339 + 0.940i)17-s + (0.269 − 0.962i)18-s + (−0.801 − 0.598i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0992 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0992 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $0.0992 + 0.995i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ 0.0992 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2703961102 + 0.2447713522i\)
\(L(\frac12)\) \(\approx\) \(0.2703961102 + 0.2447713522i\)
\(L(1)\) \(\approx\) \(0.6425355811 - 0.2112491838i\)
\(L(1)\) \(\approx\) \(0.6425355811 - 0.2112491838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (0.325 - 0.945i)T \)
3 \( 1 + (-0.999 + 0.0291i)T \)
5 \( 1 + (-0.400 + 0.916i)T \)
11 \( 1 + (0.177 - 0.984i)T \)
13 \( 1 + (0.0691 + 0.997i)T \)
17 \( 1 + (-0.339 + 0.940i)T \)
19 \( 1 + (-0.801 - 0.598i)T \)
23 \( 1 + (0.700 - 0.713i)T \)
29 \( 1 + (-0.967 - 0.252i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.141 + 0.989i)T \)
41 \( 1 + (-0.571 - 0.820i)T \)
43 \( 1 + (0.893 + 0.449i)T \)
47 \( 1 + (0.997 + 0.0728i)T \)
53 \( 1 + (0.809 + 0.586i)T \)
59 \( 1 + (-0.127 + 0.991i)T \)
61 \( 1 + (0.977 + 0.209i)T \)
67 \( 1 + (-0.485 - 0.874i)T \)
71 \( 1 + (-0.439 - 0.898i)T \)
73 \( 1 + (-0.646 - 0.762i)T \)
79 \( 1 + (0.992 + 0.123i)T \)
83 \( 1 + (0.559 + 0.828i)T \)
89 \( 1 + (0.674 - 0.738i)T \)
97 \( 1 + (0.695 + 0.718i)T \)
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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

−17.42610102893751074123176485243, −16.86353492261711164790999477892, −16.2207075503076628492972583130, −15.728081236597832054118824200986, −15.11624027625158125901141406898, −14.52456506461253818355257184868, −13.340864954743910952807052321152, −12.9124400255623179772758419894, −12.506606753888632686553907300128, −11.81919981014088617667153733768, −11.105472818466284513008366933194, −10.16605534236601691335597264356, −9.411425272486708693163703193, −8.86684423482116206497809585917, −7.92299270211836718163054791082, −7.29042913914542322748252911954, −6.91848164369965486634169009757, −5.74444608154197458003604067237, −5.44910074505379371795782123626, −4.786517488155582105259377162997, −4.1245636276005787893958718056, −3.50396898537632641911514212374, −2.10619041188493873254429779664, −1.03590711993984868799613447640, −0.13572644191294885369485238158, 0.85137101225174286637146948054, 1.8791373833908766684811770064, 2.54537065637921549614124581879, 3.628825202303089494058136020875, 4.03990598760055913590074369506, 4.72247230466952791621789036650, 5.71766929661767548851029503705, 6.26586488453610250287483201694, 6.79930163620614166434488211722, 7.74919810992131731612282117781, 8.8398982135509863111473501349, 9.24882034384577002923740651925, 10.46042532733797648745038121140, 10.68901424679916770288578989023, 11.20254303751964849206987424896, 11.793391668945516664985090977018, 12.37037949165039309429241978895, 13.2854467933492870428105017838, 13.62322771507689912081571763757, 14.7692325779954750726247057712, 14.94900757697704874137782061454, 15.862683126280595959746692682219, 16.77806331589915961497557465601, 17.17705636924265160462054365375, 18.07796407134813067413608049642

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