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Properties

Label 1-6039-6039.4084-r0-0-0
Degree $1$
Conductor $6039$
Sign $-0.637 + 0.770i$
Analytic cond. $28.0449$
Root an. cond. $28.0449$
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  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.104 − 0.994i)5-s + (−0.5 − 0.866i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)10-s + (0.669 + 0.743i)13-s + (−0.104 − 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.809 + 0.587i)17-s + 19-s + (0.669 − 0.743i)20-s + (−0.978 − 0.207i)23-s + (−0.978 + 0.207i)25-s + (0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.104 − 0.994i)5-s + (−0.5 − 0.866i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)10-s + (0.669 + 0.743i)13-s + (−0.104 − 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.809 + 0.587i)17-s + 19-s + (0.669 − 0.743i)20-s + (−0.978 − 0.207i)23-s + (−0.978 + 0.207i)25-s + (0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $-0.637 + 0.770i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (4084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ -0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6909357281 + 1.468982311i\)
\(L(\frac12)\) \(\approx\) \(0.6909357281 + 1.468982311i\)
\(L(1)\) \(\approx\) \(1.438776402 + 0.3275312443i\)
\(L(1)\) \(\approx\) \(1.438776402 + 0.3275312443i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.978 + 0.207i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−17.82419459751752538362718767405, −16.50276097956534517203296089766, −15.770662853815355238981859004226, −15.48413060188633993043236729074, −14.90648019819377885937185750860, −14.031555907527846276743151144066, −13.6109485909723354468911329602, −12.92483392043746830510580786291, −12.127204857047938525747074297945, −11.59832592535041166733902974049, −11.03090496623187591990800522030, −10.3281081488803396302037818637, −9.66261968237287926356893851404, −9.014938425129734106942330844635, −7.8479365078070833675382867725, −7.268400386798793946443635520509, −6.39548848013233145559218243207, −5.90112895684767229817779672937, −5.39311522900826282212096034335, −4.316842799765774150862289036065, −3.60134237824337236360660157792, −2.9523391029332869000897805864, −2.44436577907771094829988867758, −1.6156435376245841025639006595, −0.26427933171578403857489645199, 1.2122143334676001386027653784, 1.85435207887880939670019465734, 3.006733211343386605488550481464, 3.82637103768243717604633525002, 4.24019282508065596571907006241, 4.88828578435332449268487847337, 5.82012755122173225715993431446, 6.31632427936412697837490790530, 7.12150971372367996584660856860, 7.76435125610905948512355490528, 8.46602465224857527664770280435, 9.20032617529032805125327045010, 9.92259997615136958325233580154, 11.01678181891155385901129539322, 11.34961811872005915449214273406, 12.281473854542132565006042926552, 12.85115986510296898443408603949, 13.31257763940374962648132084271, 13.968087666622591768283126922763, 14.46734095234359618637842404433, 15.520124794702478744704193151045, 15.99233305258426657157863411227, 16.55944483283388501303993848133, 16.873634719231265010797716266126, 17.782475014251211661455744881280

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