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Properties

Label 1-33e2-1089.2-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.576 + 0.816i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.449 + 0.893i)2-s + (−0.595 + 0.803i)4-s + (−0.999 − 0.0380i)5-s + (−0.123 − 0.992i)7-s + (−0.985 − 0.170i)8-s + (−0.415 − 0.909i)10-s + (−0.00951 + 0.999i)13-s + (0.830 − 0.556i)14-s + (−0.290 − 0.956i)16-s + (0.941 − 0.336i)17-s + (0.0285 − 0.999i)19-s + (0.625 − 0.780i)20-s + (−0.981 + 0.189i)23-s + (0.997 + 0.0760i)25-s + (−0.897 + 0.441i)26-s + ⋯
L(s)  = 1  + (0.449 + 0.893i)2-s + (−0.595 + 0.803i)4-s + (−0.999 − 0.0380i)5-s + (−0.123 − 0.992i)7-s + (−0.985 − 0.170i)8-s + (−0.415 − 0.909i)10-s + (−0.00951 + 0.999i)13-s + (0.830 − 0.556i)14-s + (−0.290 − 0.956i)16-s + (0.941 − 0.336i)17-s + (0.0285 − 0.999i)19-s + (0.625 − 0.780i)20-s + (−0.981 + 0.189i)23-s + (0.997 + 0.0760i)25-s + (−0.897 + 0.441i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.576 + 0.816i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.576 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5028584070 + 0.9705918796i\)
\(L(\frac12)\) \(\approx\) \(0.5028584070 + 0.9705918796i\)
\(L(1)\) \(\approx\) \(0.8375079886 + 0.4889181345i\)
\(L(1)\) \(\approx\) \(0.8375079886 + 0.4889181345i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.449 + 0.893i)T \)
5 \( 1 + (-0.999 - 0.0380i)T \)
7 \( 1 + (-0.123 - 0.992i)T \)
13 \( 1 + (-0.00951 + 0.999i)T \)
17 \( 1 + (0.941 - 0.336i)T \)
19 \( 1 + (0.0285 - 0.999i)T \)
23 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (-0.432 + 0.901i)T \)
31 \( 1 + (0.749 + 0.662i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (0.964 + 0.263i)T \)
43 \( 1 + (0.786 - 0.618i)T \)
47 \( 1 + (0.935 + 0.353i)T \)
53 \( 1 + (-0.974 + 0.226i)T \)
59 \( 1 + (0.710 + 0.703i)T \)
61 \( 1 + (-0.548 - 0.836i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (-0.610 + 0.791i)T \)
73 \( 1 + (-0.516 + 0.856i)T \)
79 \( 1 + (0.969 + 0.244i)T \)
83 \( 1 + (-0.905 + 0.424i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.999 - 0.0380i)T \)
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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

−20.95250856621527568406265426212, −20.59011969727699436912129983351, −19.4799707469417153006502858171, −19.086089309575411261433445291298, −18.38730754354733630320746394685, −17.5463543400634588762238962231, −16.25617035609217012443405331481, −15.4945372477857306894623965431, −14.86931249626464741353943409707, −14.15227278682909440919026930568, −12.9798788822418530076524896672, −12.225998730777543186674534619978, −11.98216084284335469957968558366, −10.92901521156868995492055188218, −10.17340308788166475597866396199, −9.31754973257295357931446926982, −8.26152536800959042001122018750, −7.72433194870339234066417949165, −6.058616622398499564207052117664, −5.64235097294179569382310450557, −4.47061610257909871033560303946, −3.63370993544346015762823485936, −2.897675005373461254702200141599, −1.87515835761290961965799389972, −0.48849963710553555689586203424, 0.99734948367327140898852402197, 2.86973136251669167037184046607, 3.802860218599065946153313458009, 4.37934121574304986011640038485, 5.23780562841944752348868851314, 6.49124847472868752914822591769, 7.1820098100732391082283562513, 7.71625899864828545941731620057, 8.6669891775462964416264739033, 9.51435783761190642760026239307, 10.65866732086247081629441531821, 11.65821606385320827024115880421, 12.27550559135880217937809185165, 13.21619237238256169848776105748, 14.120898102645389557426663697682, 14.48151242582930121321994003827, 15.76621781187448742169899236023, 16.0170091980694698207703438090, 16.88165941621545434251582873286, 17.53703781660043754732157547786, 18.61422533775022838538380144772, 19.31686323311255707630387545053, 20.22566416776035376813811778772, 20.96918699874078377444240518656, 21.94598781704239354262569131684

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