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Properties

Label 1-6027-6027.3065-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.974 - 0.225i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.936 − 0.351i)2-s + (0.753 + 0.657i)4-s + (0.983 − 0.178i)5-s + (−0.473 − 0.880i)8-s + (−0.983 − 0.178i)10-s + (−0.963 + 0.266i)11-s + (0.936 + 0.351i)13-s + (0.134 + 0.990i)16-s + (−0.753 + 0.657i)17-s + (0.309 − 0.951i)19-s + (0.858 + 0.512i)20-s + (0.995 + 0.0896i)22-s + (−0.858 + 0.512i)23-s + (0.936 − 0.351i)25-s + (−0.753 − 0.657i)26-s + ⋯
L(s)  = 1  + (−0.936 − 0.351i)2-s + (0.753 + 0.657i)4-s + (0.983 − 0.178i)5-s + (−0.473 − 0.880i)8-s + (−0.983 − 0.178i)10-s + (−0.963 + 0.266i)11-s + (0.936 + 0.351i)13-s + (0.134 + 0.990i)16-s + (−0.753 + 0.657i)17-s + (0.309 − 0.951i)19-s + (0.858 + 0.512i)20-s + (0.995 + 0.0896i)22-s + (−0.858 + 0.512i)23-s + (0.936 − 0.351i)25-s + (−0.753 − 0.657i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.974 - 0.225i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (3065, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.974 - 0.225i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04015166718 - 0.3520617252i\)
\(L(\frac12)\) \(\approx\) \(0.04015166718 - 0.3520617252i\)
\(L(1)\) \(\approx\) \(0.6888467005 - 0.1275131208i\)
\(L(1)\) \(\approx\) \(0.6888467005 - 0.1275131208i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.936 - 0.351i)T \)
5 \( 1 + (0.983 - 0.178i)T \)
11 \( 1 + (-0.963 + 0.266i)T \)
13 \( 1 + (0.936 + 0.351i)T \)
17 \( 1 + (-0.753 + 0.657i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.858 + 0.512i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.393 - 0.919i)T \)
43 \( 1 + (-0.691 - 0.722i)T \)
47 \( 1 + (-0.936 - 0.351i)T \)
53 \( 1 + (0.753 + 0.657i)T \)
59 \( 1 + (-0.691 - 0.722i)T \)
61 \( 1 + (-0.858 - 0.512i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 - T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.936 + 0.351i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−18.07464692330569917073093791591, −17.5625972635442137011099425971, −16.57929263940958512296410763322, −16.27331436533533211882366794513, −15.55520336397928844093320826723, −14.86760487728397001750574952396, −14.10744715924905492821143995441, −13.52077213483922318732530719802, −12.91787793364965990863564301258, −11.87076076098711762957356225860, −11.18828266430493053002818361182, −10.45221783324007958681761647000, −10.105868945232899145019007342587, −9.407977007246863794616161711626, −8.55532809888021097552424001791, −8.17684533143932305170887040697, −7.31003498680367863537193396849, −6.52279772645978214345319118849, −5.96945843206393749208187370959, −5.42087225431194397340463407142, −4.59158038886767104946324547437, −3.231038985256484442789112447835, −2.66838509511266872317831094247, −1.795966509454928628264459179248, −1.14662388662427968549208544409, 0.11653065153677499806692347634, 1.30006781063365076088515120551, 1.97864562687005067598344493176, 2.52895208796153440996044223947, 3.44069307676061956669711171857, 4.322690920879162454056019708437, 5.268071257571468952034107185650, 6.081423978117934086829262891012, 6.62560392794164942428460398071, 7.42153488572322018189658684123, 8.27449231992925980843235381519, 8.7533080968062869847202748035, 9.48061202740410696010070367262, 10.04651493723490219357122019919, 10.70638156527184249184017285567, 11.21179807118840484098913545920, 12.061090072289936586911285007036, 12.80480696752332953519867780383, 13.46101595602694673037408284286, 13.80416452851556122224528848194, 15.04232865280977718725359374100, 15.715270190305818086706550981694, 16.0609760539587590686567537588, 17.09323673365960920831042115809, 17.409048679321728611790279778106

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