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Properties

Label 1-33e2-1089.1040-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.941 + 0.336i$
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  + (−0.969 + 0.244i)2-s + (0.879 − 0.475i)4-s + (−0.861 − 0.508i)5-s + (−0.161 − 0.986i)7-s + (−0.736 + 0.676i)8-s + (0.959 + 0.281i)10-s + (0.991 + 0.132i)13-s + (0.398 + 0.917i)14-s + (0.548 − 0.836i)16-s + (0.0855 + 0.996i)17-s + (0.921 + 0.389i)19-s + (−0.999 − 0.0380i)20-s + (0.888 + 0.458i)23-s + (0.483 + 0.875i)25-s + (−0.993 + 0.113i)26-s + ⋯
L(s)  = 1  + (−0.969 + 0.244i)2-s + (0.879 − 0.475i)4-s + (−0.861 − 0.508i)5-s + (−0.161 − 0.986i)7-s + (−0.736 + 0.676i)8-s + (0.959 + 0.281i)10-s + (0.991 + 0.132i)13-s + (0.398 + 0.917i)14-s + (0.548 − 0.836i)16-s + (0.0855 + 0.996i)17-s + (0.921 + 0.389i)19-s + (−0.999 − 0.0380i)20-s + (0.888 + 0.458i)23-s + (0.483 + 0.875i)25-s + (−0.993 + 0.113i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1040, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7485401890 + 0.1297623983i\)
\(L(\frac12)\) \(\approx\) \(0.7485401890 + 0.1297623983i\)
\(L(1)\) \(\approx\) \(0.6444372710 + 0.003567305562i\)
\(L(1)\) \(\approx\) \(0.6444372710 + 0.003567305562i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.969 + 0.244i)T \)
5 \( 1 + (-0.861 - 0.508i)T \)
7 \( 1 + (-0.161 - 0.986i)T \)
13 \( 1 + (0.991 + 0.132i)T \)
17 \( 1 + (0.0855 + 0.996i)T \)
19 \( 1 + (0.921 + 0.389i)T \)
23 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (-0.999 + 0.0190i)T \)
31 \( 1 + (-0.761 - 0.647i)T \)
37 \( 1 + (-0.564 + 0.825i)T \)
41 \( 1 + (-0.830 - 0.556i)T \)
43 \( 1 + (0.995 + 0.0950i)T \)
47 \( 1 + (-0.345 + 0.938i)T \)
53 \( 1 + (0.998 - 0.0570i)T \)
59 \( 1 + (0.0665 + 0.997i)T \)
61 \( 1 + (-0.272 - 0.962i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.974 + 0.226i)T \)
73 \( 1 + (0.254 + 0.967i)T \)
79 \( 1 + (0.948 + 0.318i)T \)
83 \( 1 + (0.988 + 0.151i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.861 - 0.508i)T \)
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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

−21.16167905646704931956238087156, −20.48977217791160611774884823604, −19.68692477566846084757184116720, −18.942532614360572670387546637545, −18.28238421759799288987455650181, −17.9842246123162850198899962430, −16.55384511945342108328755609254, −16.03455498961893549456960716748, −15.37737369682978175932456626289, −14.67523466837825908612490812327, −13.37545129209141197469670889000, −12.38089309861979063784474311301, −11.671798581371315750584331684816, −11.12685355584252071831128604187, −10.30943466389707770010428918122, −9.086544091253334233532440097657, −8.82555083923867613875796945605, −7.67286993275605803772499236086, −7.08899183059245488399965790537, −6.14082122039052443836494642316, −5.06728952064583217099902191130, −3.527695016220296634712835730957, −3.05654975978979193180697532449, −1.98037027242530043830129248982, −0.60379794244987847497264159525, 0.893626811909202466025843030570, 1.61076252701639656248045661341, 3.314930922205073837643298013671, 3.935434207393000206797267609115, 5.236210149718542027486112245470, 6.19545277115070375359011694231, 7.26104051007441159716385890624, 7.71529163588799649089716527031, 8.62113660755213855639818106787, 9.32546124948812738340013096551, 10.35189570367299511172019313701, 11.05305740827543548863110072365, 11.69238627665985964487234401156, 12.764883353200363243176388676511, 13.602550554288401034392847666056, 14.71631677110910123144358932173, 15.45793210965484765277745459697, 16.22781479661553203398933140490, 16.786919476106274769982047346407, 17.427737110526002027748010642952, 18.50817290753137682914606175134, 19.13028422614878150866860414513, 19.82584010746969655224446794468, 20.60306173034512428266291194515, 20.92739776025209919012380401581

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