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Properties

Label 1-6041-6041.160-r0-0-0
Degree $1$
Conductor $6041$
Sign $-0.767 + 0.640i$
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  + (−0.967 − 0.252i)2-s + (−0.0982 + 0.995i)3-s + (0.872 + 0.488i)4-s + (0.917 − 0.396i)5-s + (0.346 − 0.938i)6-s + (−0.721 − 0.692i)8-s + (−0.980 − 0.195i)9-s + (−0.988 + 0.152i)10-s + (−0.929 − 0.370i)11-s + (−0.571 + 0.820i)12-s + (−0.908 − 0.416i)13-s + (0.304 + 0.952i)15-s + (0.522 + 0.852i)16-s + (−0.547 − 0.836i)17-s + (0.899 + 0.436i)18-s + (−0.559 − 0.828i)19-s + ⋯
L(s)  = 1  + (−0.967 − 0.252i)2-s + (−0.0982 + 0.995i)3-s + (0.872 + 0.488i)4-s + (0.917 − 0.396i)5-s + (0.346 − 0.938i)6-s + (−0.721 − 0.692i)8-s + (−0.980 − 0.195i)9-s + (−0.988 + 0.152i)10-s + (−0.929 − 0.370i)11-s + (−0.571 + 0.820i)12-s + (−0.908 − 0.416i)13-s + (0.304 + 0.952i)15-s + (0.522 + 0.852i)16-s + (−0.547 − 0.836i)17-s + (0.899 + 0.436i)18-s + (−0.559 − 0.828i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $-0.767 + 0.640i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ -0.767 + 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03102649977 + 0.08560103016i\)
\(L(\frac12)\) \(\approx\) \(0.03102649977 + 0.08560103016i\)
\(L(1)\) \(\approx\) \(0.5810390667 + 0.005140002970i\)
\(L(1)\) \(\approx\) \(0.5810390667 + 0.005140002970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (-0.967 - 0.252i)T \)
3 \( 1 + (-0.0982 + 0.995i)T \)
5 \( 1 + (0.917 - 0.396i)T \)
11 \( 1 + (-0.929 - 0.370i)T \)
13 \( 1 + (-0.908 - 0.416i)T \)
17 \( 1 + (-0.547 - 0.836i)T \)
19 \( 1 + (-0.559 - 0.828i)T \)
23 \( 1 + (0.657 - 0.753i)T \)
29 \( 1 + (0.311 - 0.950i)T \)
31 \( 1 + (-0.380 + 0.924i)T \)
37 \( 1 + (-0.994 + 0.109i)T \)
41 \( 1 + (0.105 + 0.994i)T \)
43 \( 1 + (-0.00364 - 0.999i)T \)
47 \( 1 + (0.858 + 0.513i)T \)
53 \( 1 + (-0.971 - 0.238i)T \)
59 \( 1 + (-0.809 + 0.586i)T \)
61 \( 1 + (-0.949 + 0.315i)T \)
67 \( 1 + (-0.944 - 0.329i)T \)
71 \( 1 + (0.227 + 0.973i)T \)
73 \( 1 + (-0.177 - 0.984i)T \)
79 \( 1 + (-0.0255 - 0.999i)T \)
83 \( 1 + (-0.589 + 0.807i)T \)
89 \( 1 + (0.433 - 0.901i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−17.50855153431552678988687664660, −17.08187605515126555925669429555, −16.5022230705129518180256592258, −15.414961301243964654891502695331, −14.8866422284302272421937874293, −14.21953993667916018401565141144, −13.584856086434156589306698636664, −12.62237135388526194932992315874, −12.39716240030968672299106541317, −11.25794461300041089202805014272, −10.748467470990386440319523138740, −10.17313232282828472729546862784, −9.36846776135637704102441683406, −8.77713823634649759228950972907, −7.96108710222092065728521382640, −7.319251023896085271735943513083, −6.869644865343786693949890495687, −6.0803059654556652522253341187, −5.59674010844788830779800550485, −4.80825451860979942113196511078, −3.276719370773554628205743488848, −2.4917369216546101573696015318, −1.8941453442439598690540175817, −1.46289637551850172267843359638, −0.03800925964987871786569836950, 0.75671761315720676870521171894, 2.05791580948324714342299033849, 2.73567984211226281859151825093, 3.10118383915013539501222449567, 4.534386475387668951313081822098, 4.93210155912822340660972034964, 5.776040084880568539062382872035, 6.49836973196509437444970441969, 7.33057814451655243865865959059, 8.207302099754857786753661896011, 8.99142376095805729188784097059, 9.17115701269407504244666418822, 10.11726560157563948467605791933, 10.480972618297479935356244630543, 10.99816212159540134085007580305, 11.88840000614589247923500025521, 12.57395303734547016596250804323, 13.31405998450018146478645687156, 14.05400355887122308684624553448, 14.970979449898441437292110909654, 15.56008026977301794845910583853, 16.095857874627285813723610247330, 16.79722096865377741156910082573, 17.3221239232261588475966768264, 17.752225821115307080694687435869

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