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Properties

Label 1-33e2-1089.1013-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.00576 - 0.999i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (0.786 + 0.618i)2-s + (0.235 + 0.971i)4-s + (−0.580 + 0.814i)5-s + (0.0475 + 0.998i)7-s + (−0.415 + 0.909i)8-s + (−0.959 + 0.281i)10-s + (0.235 + 0.971i)13-s + (−0.580 + 0.814i)14-s + (−0.888 + 0.458i)16-s + (0.654 − 0.755i)17-s + (−0.654 − 0.755i)19-s + (−0.928 − 0.371i)20-s + (−0.0475 + 0.998i)23-s + (−0.327 − 0.945i)25-s + (−0.415 + 0.909i)26-s + ⋯
L(s)  = 1  + (0.786 + 0.618i)2-s + (0.235 + 0.971i)4-s + (−0.580 + 0.814i)5-s + (0.0475 + 0.998i)7-s + (−0.415 + 0.909i)8-s + (−0.959 + 0.281i)10-s + (0.235 + 0.971i)13-s + (−0.580 + 0.814i)14-s + (−0.888 + 0.458i)16-s + (0.654 − 0.755i)17-s + (−0.654 − 0.755i)19-s + (−0.928 − 0.371i)20-s + (−0.0475 + 0.998i)23-s + (−0.327 − 0.945i)25-s + (−0.415 + 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00576 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00576 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.00576 - 0.999i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.00576 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9719558075 + 0.9663640460i\)
\(L(\frac12)\) \(\approx\) \(-0.9719558075 + 0.9663640460i\)
\(L(1)\) \(\approx\) \(0.8306955209 + 0.9626625165i\)
\(L(1)\) \(\approx\) \(0.8306955209 + 0.9626625165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.786 + 0.618i)T \)
5 \( 1 + (-0.580 + 0.814i)T \)
7 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (0.654 - 0.755i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (-0.981 + 0.189i)T \)
31 \( 1 + (0.723 + 0.690i)T \)
37 \( 1 + (-0.959 + 0.281i)T \)
41 \( 1 + (-0.928 + 0.371i)T \)
43 \( 1 + (0.580 + 0.814i)T \)
47 \( 1 + (-0.928 - 0.371i)T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (0.786 - 0.618i)T \)
61 \( 1 + (0.928 + 0.371i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (-0.995 - 0.0950i)T \)
83 \( 1 + (-0.0475 - 0.998i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (0.580 + 0.814i)T \)
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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

−20.77487009276355695770414593024, −20.11717519703002412602028826355, −19.30365218218749333778254891345, −18.72231950535528781095609605958, −17.389112882493764048714529938, −16.72440227672965748302034938702, −15.86385284012432017738658818930, −15.035461637515569356155175913225, −14.32528880114616660619893769814, −13.340366413784001035119147499876, −12.756041337970501467786828041974, −12.16823320866181479702519919823, −11.138122958597742052185374886462, −10.45278208056793349591192062875, −9.7797978841448363240330988528, −8.47591915985049888447751197761, −7.8084107174801987483824050295, −6.66925234753981911684362927123, −5.67963968510087351547186707614, −4.87098100006321074422014452894, −3.878336726119144539947286161464, −3.54995767589362660169775957916, −2.04228181975255805875727906272, −1.02305737557191869251960483225, −0.22205762684198489668481626295, 1.90959367846187378851910079348, 2.905449202884657936273541785248, 3.59184999460219728243782123873, 4.671158550406030650390354014753, 5.47343080045285949510731427527, 6.50856298319550360280280674209, 7.01666637488952019230841685594, 8.002563409594352371772496121, 8.77399866005572701558547698733, 9.74851689796894916202167318592, 11.22287433439554995076857754268, 11.55850571697534093837939620521, 12.33518259435515700807680742887, 13.302718003276706522925928040005, 14.21927208738702965901886582601, 14.73149595696805631132834603518, 15.65342531890611895209043874574, 15.93576671986010036129229099870, 17.04599479741046549518853115206, 17.91792862885235362941846452780, 18.73455907407541302551558236245, 19.342756244696229782920657012517, 20.48278783100136645674103333740, 21.48928644763699388386930232954, 21.75339203163312142156632095351

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