↑

Properties

Label 1-6045-6045.2813-r0-0-0
Degree $1$
Conductor $6045$
Sign $0.589 - 0.807i$
Analytic cond. $28.0728$
Root an. cond. $28.0728$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.951 + 0.309i)11-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s + (0.587 − 0.809i)19-s + (0.951 + 0.309i)22-s + (−0.951 − 0.309i)23-s + (−0.809 + 0.587i)28-s + (0.809 + 0.587i)29-s + 32-s + (−0.587 − 0.809i)34-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.951 + 0.309i)11-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s + (0.587 − 0.809i)19-s + (0.951 + 0.309i)22-s + (−0.951 − 0.309i)23-s + (−0.809 + 0.587i)28-s + (0.809 + 0.587i)29-s + 32-s + (−0.587 − 0.809i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $0.589 - 0.807i$
Analytic conductor: \(28.0728\)
Root analytic conductor: \(28.0728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6045} (2813, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6045,\ (0:\ ),\ 0.589 - 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9113870211 - 0.4632091433i\)
\(L(\frac12)\) \(\approx\) \(0.9113870211 - 0.4632091433i\)
\(L(1)\) \(\approx\) \(0.7302562140 - 0.1130037235i\)
\(L(1)\) \(\approx\) \(0.7302562140 - 0.1130037235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 - T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.96502857805998237709331051239, −17.07887132547925692219832346312, −16.4704270926961572673588402632, −16.05997866904656321652490839434, −15.362309930816456566351681640915, −14.545096810161447601069090111295, −13.86073914329226728101151834753, −13.65301659180878707790442864124, −12.42789996807572479699387559725, −11.75420450264416725506510195587, −10.9266702300490241060159736320, −10.33956625570124298123905844418, −9.92896463125893646143672619676, −9.16836030824528246516435707680, −8.13251810129796155699134550580, −7.80493752501154455035393267764, −7.3572629219131246432000282599, −6.35262009262016926750268862228, −5.73800101015811445366701138645, −5.05863140336948772755639806227, −4.253184888466832374599531136959, −3.28332354713535788093752957652, −2.40318762199709577050704605999, −1.38593683387948571994075848473, −0.77703063398721425336978920035, 0.456051611812286689803342996000, 1.54238979348031194144848232082, 2.280890243071798601299400946230, 2.849960423934571627511093806580, 3.632815213534612511157064003070, 4.62649755384566430370872148408, 5.3769149470213292262471414467, 6.09186468847428155875646812326, 7.18003684020501551263777199401, 7.618945590834867654275591399932, 8.46718495118467387349860706605, 8.85286017548112602469683866696, 9.687027216193042099171535997510, 10.35030933477367702826382380862, 10.82402187674230930680366082149, 11.73636201816685980210803335360, 12.294648417053764015207030892, 12.602563111127978838453234856886, 13.63111031022581567310638752657, 14.2367834016709255222769477433, 15.335860400582283145681233762248, 15.6747204706628181484677749121, 16.29991138892263561711112171757, 17.1758138378773956808613637227, 17.777672952575208066169209600129

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