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Properties

Label 1-33e2-1089.205-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.819 + 0.573i$
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  + (−0.953 − 0.299i)2-s + (0.820 + 0.572i)4-s + (−0.851 − 0.524i)5-s + (0.532 − 0.846i)7-s + (−0.610 − 0.791i)8-s + (0.654 + 0.755i)10-s + (−0.797 + 0.603i)13-s + (−0.761 + 0.647i)14-s + (0.345 + 0.938i)16-s + (0.254 + 0.967i)17-s + (0.362 + 0.931i)19-s + (−0.398 − 0.917i)20-s + (0.928 + 0.371i)23-s + (0.449 + 0.893i)25-s + (0.941 − 0.336i)26-s + ⋯
L(s)  = 1  + (−0.953 − 0.299i)2-s + (0.820 + 0.572i)4-s + (−0.851 − 0.524i)5-s + (0.532 − 0.846i)7-s + (−0.610 − 0.791i)8-s + (0.654 + 0.755i)10-s + (−0.797 + 0.603i)13-s + (−0.761 + 0.647i)14-s + (0.345 + 0.938i)16-s + (0.254 + 0.967i)17-s + (0.362 + 0.931i)19-s + (−0.398 − 0.917i)20-s + (0.928 + 0.371i)23-s + (0.449 + 0.893i)25-s + (0.941 − 0.336i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.819 + 0.573i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.819 + 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01005757745 + 0.03190658401i\)
\(L(\frac12)\) \(\approx\) \(0.01005757745 + 0.03190658401i\)
\(L(1)\) \(\approx\) \(0.5666969640 - 0.1122708865i\)
\(L(1)\) \(\approx\) \(0.5666969640 - 0.1122708865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.953 - 0.299i)T \)
5 \( 1 + (-0.851 - 0.524i)T \)
7 \( 1 + (0.532 - 0.846i)T \)
13 \( 1 + (-0.797 + 0.603i)T \)
17 \( 1 + (0.254 + 0.967i)T \)
19 \( 1 + (0.362 + 0.931i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (-0.548 - 0.836i)T \)
31 \( 1 + (0.483 + 0.875i)T \)
37 \( 1 + (0.974 + 0.226i)T \)
41 \( 1 + (-0.749 - 0.662i)T \)
43 \( 1 + (-0.235 - 0.971i)T \)
47 \( 1 + (0.861 + 0.508i)T \)
53 \( 1 + (-0.985 + 0.170i)T \)
59 \( 1 + (-0.948 - 0.318i)T \)
61 \( 1 + (0.217 - 0.976i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (0.774 - 0.633i)T \)
73 \( 1 + (-0.696 - 0.717i)T \)
79 \( 1 + (0.432 - 0.901i)T \)
83 \( 1 + (0.0665 + 0.997i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.851 + 0.524i)T \)
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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

−20.62023406308581174907814430649, −20.00771769187753464950579541579, −19.232481557046326467036055864263, −18.45222864067628152927979142249, −18.079999536613972907487410806718, −17.09581944247156504750793708659, −16.257735030838010664051435882428, −15.39159041408655864168017318696, −14.987110763526762132323012537932, −14.30034341419760940013341114220, −12.87060927400746788166937226502, −11.79204035053589505350218831438, −11.43123935427046737511399116290, −10.586169859890679754262736159891, −9.56775334589517224958410130704, −8.871234801583210285104403907450, −7.905133845537637694857962408041, −7.403704427893464145182100424022, −6.54939290060279163696897292883, −5.40507806861053054981964039737, −4.67997141760468409856834420916, −2.94387078866671722435740015372, −2.577301267675320770565031480393, −1.06541225399822658376851053406, −0.01169735329208511985134896338, 1.08075179258815219366014466314, 1.85167679944376687160737754355, 3.30360273590202780505575766052, 4.042681414136355375477221570574, 5.00198656820412907115607455329, 6.38181394081213414404379986427, 7.46605793838740781471187796446, 7.78193407594970411849049043888, 8.68372787906959864153971598364, 9.550619130923126204439331766626, 10.44532381761000800249295232846, 11.15275583102113013798545174898, 11.983493407073810212137623966334, 12.515847764971814485495288566428, 13.6243642126620767773781625843, 14.7364701741356636296667651876, 15.4316554125009309467150983023, 16.42596111171798079132967004915, 17.01372132917056096903417185624, 17.42212320212203939462025294861, 18.7330917181773190193397739733, 19.180326093423533891853685515226, 19.95058031589509892309368835560, 20.61067559890745632570027827988, 21.19157299405911800418528059546

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