↑

Properties

Label 1-33e2-1089.4-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.956 + 0.292i$
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  + (−0.595 + 0.803i)2-s + (−0.290 − 0.956i)4-s + (0.997 + 0.0760i)5-s + (−0.969 + 0.244i)7-s + (0.941 + 0.336i)8-s + (−0.654 + 0.755i)10-s + (−0.999 − 0.0190i)13-s + (0.380 − 0.924i)14-s + (−0.830 + 0.556i)16-s + (0.774 − 0.633i)17-s + (−0.998 − 0.0570i)19-s + (−0.217 − 0.976i)20-s + (0.928 − 0.371i)23-s + (0.988 + 0.151i)25-s + (0.610 − 0.791i)26-s + ⋯
L(s)  = 1  + (−0.595 + 0.803i)2-s + (−0.290 − 0.956i)4-s + (0.997 + 0.0760i)5-s + (−0.969 + 0.244i)7-s + (0.941 + 0.336i)8-s + (−0.654 + 0.755i)10-s + (−0.999 − 0.0190i)13-s + (0.380 − 0.924i)14-s + (−0.830 + 0.556i)16-s + (0.774 − 0.633i)17-s + (−0.998 − 0.0570i)19-s + (−0.217 − 0.976i)20-s + (0.928 − 0.371i)23-s + (0.988 + 0.151i)25-s + (0.610 − 0.791i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.956 + 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014321314 + 0.1515165490i\)
\(L(\frac12)\) \(\approx\) \(1.014321314 + 0.1515165490i\)
\(L(1)\) \(\approx\) \(0.7933989671 + 0.2090362900i\)
\(L(1)\) \(\approx\) \(0.7933989671 + 0.2090362900i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.595 + 0.803i)T \)
5 \( 1 + (0.997 + 0.0760i)T \)
7 \( 1 + (-0.969 + 0.244i)T \)
13 \( 1 + (-0.999 - 0.0190i)T \)
17 \( 1 + (0.774 - 0.633i)T \)
19 \( 1 + (-0.998 - 0.0570i)T \)
23 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (-0.625 - 0.780i)T \)
31 \( 1 + (0.123 + 0.992i)T \)
37 \( 1 + (0.0855 - 0.996i)T \)
41 \( 1 + (0.861 + 0.508i)T \)
43 \( 1 + (0.235 - 0.971i)T \)
47 \( 1 + (0.749 + 0.662i)T \)
53 \( 1 + (0.897 - 0.441i)T \)
59 \( 1 + (0.00951 + 0.999i)T \)
61 \( 1 + (-0.398 + 0.917i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (-0.254 - 0.967i)T \)
73 \( 1 + (-0.466 - 0.884i)T \)
79 \( 1 + (0.879 + 0.475i)T \)
83 \( 1 + (0.640 - 0.768i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.997 - 0.0760i)T \)
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

−21.38051122940325206911995507956, −20.59736808811796943030755916723, −19.79065383666241438769019021113, −19.04448030108632405273998787102, −18.54903877736209642171792916031, −17.285706347015083702525225895920, −17.07366721013192945577524645435, −16.38115014289425293383309686837, −15.08688733664802364137382825441, −14.18537585300329750915263214153, −13.07831213925529197966422257864, −12.88286576254069484891910792334, −11.96624048073698199224079938043, −10.77329806587275372430527980022, −10.2114784549260730503638054549, −9.498426560147768822803587403241, −8.94330401055355588916940506618, −7.7681556978548006487039821475, −6.90870686566472024862546972420, −5.97163826576400213135276082457, −4.868414726712485490116295714705, −3.76847681745869469830710562371, −2.83301157051263266210627256735, −2.05181926747060414536811754590, −0.91923773511931118439656326967, 0.66257250457536621926113443112, 2.064638884422794793779489042688, 2.8943305311974802620233548511, 4.44655066642582314012497906431, 5.44136820221587871935014203457, 6.04222361576057478758985733992, 6.91609121091449627559502001728, 7.554251362384241462193968466139, 8.96301797882727224171062613826, 9.24986306994473278305171269636, 10.17821275054244566157835789012, 10.6679812447766232083286161543, 12.13207690072785474305662222687, 12.99083244466512452814879302467, 13.74053690219250149010024843475, 14.62949127255294410531896173577, 15.16216136100277467534066997845, 16.31632832412890073220314420836, 16.74531994645015114689127496454, 17.509026044006965055135277085292, 18.2490752565874562370884712254, 19.14530438278789444034033784037, 19.49496943077693565441473402215, 20.71901835398951098551470680670, 21.50385241378668621305940648358

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