↑

Properties

Label 1-552-552.515-r0-0-0
Degree $1$
Conductor $552$
Sign $0.986 - 0.165i$
Analytic cond. $2.56347$
Root an. cond. $2.56347$
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  + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (−0.841 − 0.540i)11-s + (0.654 − 0.755i)13-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (0.841 − 0.540i)25-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.841 − 0.540i)35-s + (0.959 + 0.281i)37-s + (0.959 − 0.281i)41-s + (−0.142 + 0.989i)43-s + 47-s + (−0.142 + 0.989i)49-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (−0.841 − 0.540i)11-s + (0.654 − 0.755i)13-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (0.841 − 0.540i)25-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.841 − 0.540i)35-s + (0.959 + 0.281i)37-s + (0.959 − 0.281i)41-s + (−0.142 + 0.989i)43-s + 47-s + (−0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(2.56347\)
Root analytic conductor: \(2.56347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 552,\ (0:\ ),\ 0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.169698161 - 0.09723219645i\)
\(L(\frac12)\) \(\approx\) \(1.169698161 - 0.09723219645i\)
\(L(1)\) \(\approx\) \(0.9797032440 + 0.002651811008i\)
\(L(1)\) \(\approx\) \(0.9797032440 + 0.002651811008i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (0.959 + 0.281i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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

−23.44695721414503316621928732547, −22.85671998501813325982887586550, −21.560265524240482956733226326831, −20.66361844182448570039985209102, −20.24950433868757754739471972451, −19.16546819644080648767650527656, −18.451732301489116178185434887062, −17.413736058581700111193066433451, −16.61476071608132053124794470889, −15.746517018023231858538394298356, −15.00180531432157408837419766370, −14.01961960593735935004749883265, −13.1109683086970262006545064344, −12.1923311584050337346091869903, −11.266481015909305145662771528172, −10.61953596222404764341253030095, −9.49293113273124810911987900281, −8.18174251761349475813114317530, −7.84609099480305699515844530366, −6.76777356236103191069943748184, −5.50974923239891068248970562782, −4.247545936653634982273210538926, −3.95220942979753640788570377912, −2.284738001057500605696965085688, −1.01206257268510578898528950217, 0.82275849604145030892571620164, 2.58400936234075358602480036698, 3.25769896444816563607278013122, 4.67692024393421739954274531153, 5.38336823111896980343229390112, 6.63065115189532847092462530095, 7.73001292235736574297559049703, 8.34907466402496770075217065013, 9.24752999606014516050507554530, 10.75808416412708502943395152460, 11.160631499978114910017062187505, 12.08624471554287951278978969971, 13.02675627891598963418134981545, 14.0378207065329374455048961288, 15.01369751627850381280085384486, 15.78809178795700141054825722987, 16.18262174765924286655009636628, 17.9115526557021601711836988597, 18.11823056361681362347187320479, 19.101578026086403183669150971773, 20.01715813926842615405560437070, 20.78162293435070696498354669815, 21.71583648133125386282902683981, 22.50516374928926021333620900156, 23.42572056499756503912088306820

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