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Properties

Label 1-33e2-1089.394-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.601 + 0.798i$
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  + (−0.948 + 0.318i)2-s + (0.797 − 0.603i)4-s + (0.380 + 0.924i)5-s + (0.640 + 0.768i)7-s + (−0.564 + 0.825i)8-s + (−0.654 − 0.755i)10-s + (−0.290 + 0.956i)13-s + (−0.851 − 0.524i)14-s + (0.272 − 0.962i)16-s + (−0.362 + 0.931i)17-s + (0.774 − 0.633i)19-s + (0.861 + 0.508i)20-s + (0.928 + 0.371i)23-s + (−0.710 + 0.703i)25-s + (−0.0285 − 0.999i)26-s + ⋯
L(s)  = 1  + (−0.948 + 0.318i)2-s + (0.797 − 0.603i)4-s + (0.380 + 0.924i)5-s + (0.640 + 0.768i)7-s + (−0.564 + 0.825i)8-s + (−0.654 − 0.755i)10-s + (−0.290 + 0.956i)13-s + (−0.851 − 0.524i)14-s + (0.272 − 0.962i)16-s + (−0.362 + 0.931i)17-s + (0.774 − 0.633i)19-s + (0.861 + 0.508i)20-s + (0.928 + 0.371i)23-s + (−0.710 + 0.703i)25-s + (−0.0285 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.601 + 0.798i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.601 + 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4751678704 + 0.9532114613i\)
\(L(\frac12)\) \(\approx\) \(0.4751678704 + 0.9532114613i\)
\(L(1)\) \(\approx\) \(0.7168253580 + 0.4079162456i\)
\(L(1)\) \(\approx\) \(0.7168253580 + 0.4079162456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.948 + 0.318i)T \)
5 \( 1 + (0.380 + 0.924i)T \)
7 \( 1 + (0.640 + 0.768i)T \)
13 \( 1 + (-0.290 + 0.956i)T \)
17 \( 1 + (-0.362 + 0.931i)T \)
19 \( 1 + (0.774 - 0.633i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (0.964 - 0.263i)T \)
31 \( 1 + (-0.905 - 0.424i)T \)
37 \( 1 + (0.516 - 0.856i)T \)
41 \( 1 + (-0.398 + 0.917i)T \)
43 \( 1 + (0.235 + 0.971i)T \)
47 \( 1 + (-0.217 + 0.976i)T \)
53 \( 1 + (0.696 - 0.717i)T \)
59 \( 1 + (-0.595 + 0.803i)T \)
61 \( 1 + (0.749 - 0.662i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (-0.998 + 0.0570i)T \)
73 \( 1 + (0.897 - 0.441i)T \)
79 \( 1 + (-0.179 - 0.983i)T \)
83 \( 1 + (-0.532 + 0.846i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.380 - 0.924i)T \)
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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

−20.81796928349103504220157443343, −20.234031301797210419365470187512, −19.97480755410875022818614725978, −18.69906864750954025976517351634, −17.94630889423418333153155585286, −17.36596209008055692069894788298, −16.68164234383452064157952838625, −16.03319233452369024796410012688, −15.08457255420409668222742157875, −13.960839632599209551210446153270, −13.1804246242881798327395738067, −12.30218424373378328448005868246, −11.61832704377411946188392355992, −10.57267991437194384259257792442, −10.066437019221022512577200710649, −9.08354834108229672876201447535, −8.42279724007174153707727522891, −7.57949673169110765647281339184, −6.883306240819264144764182895121, −5.53277458245412926595004787013, −4.7699835863542859232161577002, −3.58129192931467663555439587641, −2.50334322008629895333722829941, −1.374276672450366421293902876098, −0.650792815872504045804428294270, 1.42387744321479004007246371404, 2.23888619367664067798441019473, 3.05410086775574123274938844093, 4.62755686104263556826317157814, 5.680932704467312832938580022662, 6.40627101714588031312528239020, 7.21677884931050711966145993356, 7.9992667732697308530150493335, 9.060329205975645061620230610633, 9.5055335987232755436213765427, 10.57942968310106649809712975021, 11.287577771133077007938842361825, 11.7899070199763311219022658864, 13.11702262574983979620924607087, 14.27850858359292813085077041474, 14.76625455656750676168229621884, 15.432147792366243683947091677865, 16.328838500864291886428332308875, 17.306619567454835996498914733297, 17.87666124245905238088763715416, 18.45821685214652949143175843921, 19.26214783194553473851300862413, 19.793670184890781799007163948081, 21.13469330998319898166412066982, 21.47191933993388858144882431648

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