Properties

Label 1-33e2-1089.173-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.799 + 0.600i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.797 + 0.603i)2-s + (0.272 + 0.962i)4-s + (0.710 + 0.703i)5-s + (0.179 + 0.983i)7-s + (−0.362 + 0.931i)8-s + (0.142 + 0.989i)10-s + (0.830 − 0.556i)13-s + (−0.449 + 0.893i)14-s + (−0.851 + 0.524i)16-s + (−0.736 + 0.676i)17-s + (−0.198 − 0.980i)19-s + (−0.483 + 0.875i)20-s + (−0.723 + 0.690i)23-s + (0.00951 + 0.999i)25-s + (0.998 + 0.0570i)26-s + ⋯
L(s)  = 1  + (0.797 + 0.603i)2-s + (0.272 + 0.962i)4-s + (0.710 + 0.703i)5-s + (0.179 + 0.983i)7-s + (−0.362 + 0.931i)8-s + (0.142 + 0.989i)10-s + (0.830 − 0.556i)13-s + (−0.449 + 0.893i)14-s + (−0.851 + 0.524i)16-s + (−0.736 + 0.676i)17-s + (−0.198 − 0.980i)19-s + (−0.483 + 0.875i)20-s + (−0.723 + 0.690i)23-s + (0.00951 + 0.999i)25-s + (0.998 + 0.0570i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.799 + 0.600i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.799 + 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8512057315 + 2.549484083i\)
\(L(\frac12)\) \(\approx\) \(0.8512057315 + 2.549484083i\)
\(L(1)\) \(\approx\) \(1.370434189 + 1.200363469i\)
\(L(1)\) \(\approx\) \(1.370434189 + 1.200363469i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.797 + 0.603i)T \)
5 \( 1 + (0.710 + 0.703i)T \)
7 \( 1 + (0.179 + 0.983i)T \)
13 \( 1 + (0.830 - 0.556i)T \)
17 \( 1 + (-0.736 + 0.676i)T \)
19 \( 1 + (-0.198 - 0.980i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (0.861 + 0.508i)T \)
31 \( 1 + (0.640 - 0.768i)T \)
37 \( 1 + (-0.466 + 0.884i)T \)
41 \( 1 + (-0.683 + 0.730i)T \)
43 \( 1 + (0.888 + 0.458i)T \)
47 \( 1 + (0.905 - 0.424i)T \)
53 \( 1 + (0.0285 - 0.999i)T \)
59 \( 1 + (0.290 - 0.956i)T \)
61 \( 1 + (-0.123 - 0.992i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.993 - 0.113i)T \)
73 \( 1 + (-0.610 - 0.791i)T \)
79 \( 1 + (0.935 + 0.353i)T \)
83 \( 1 + (-0.432 + 0.901i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.710 + 0.703i)T \)
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

−20.898031172422978886103649280314, −20.652324130758334388804794240665, −19.88751286995864575852676665415, −18.980169137695719002029651380312, −18.05708737623863327832474531495, −17.27754463795900304393645516008, −16.24127046625780782633552925317, −15.80614399214771813468205166792, −14.40581293706963042220972127222, −13.85042551331594557318832728800, −13.472368257246599057453874939844, −12.4309590666320122226510300078, −11.83016058904430286245859608384, −10.64104866440833384884504213065, −10.30704465664082051066443365894, −9.245152950241762073572807014962, −8.441197806990214646803052762845, −7.08560742330860390064459639047, −6.26473736047633866210082009148, −5.48115388609998853805444709926, −4.349556694371723473173780511804, −4.064376183420195919038960846764, −2.639937398573495850123227294786, −1.69166441369533080981153870403, −0.83623854998722449515059663872, 1.82715336891455514484106353294, 2.66947125037846788604254417826, 3.46440948144925186122002358465, 4.658128142843998610068836469587, 5.561264104826082598889210236875, 6.23899457258733902164475650172, 6.81163382567072720247369250448, 8.07516801973161712324170117880, 8.6680668681127305937416365247, 9.71404477322868501022038239987, 10.882091376286045174942621744790, 11.48236581880649089159787019433, 12.484675323289076182994029728315, 13.32156821665392007452344522541, 13.82799598748821570731195128927, 14.83552196842336669726359606878, 15.39314726949378414739711404173, 15.921614917349161444912853784118, 17.23460538430851582851285127142, 17.73239184924688241363239864149, 18.36424734015598915219246076309, 19.41312676872418108434770888905, 20.48878434985847191828800623831, 21.27782172989361523260462416348, 21.99744994110104768363238701375

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