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Properties

Label 1-33e2-1089.211-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.544 + 0.838i$
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  + (−0.749 − 0.662i)2-s + (0.123 + 0.992i)4-s + (−0.991 − 0.132i)5-s + (−0.345 + 0.938i)7-s + (0.564 − 0.825i)8-s + (0.654 + 0.755i)10-s + (0.683 + 0.730i)13-s + (0.879 − 0.475i)14-s + (−0.969 + 0.244i)16-s + (0.362 − 0.931i)17-s + (−0.774 + 0.633i)19-s + (0.00951 − 0.999i)20-s + (−0.786 + 0.618i)23-s + (0.964 + 0.263i)25-s + (−0.0285 − 0.999i)26-s + ⋯
L(s)  = 1  + (−0.749 − 0.662i)2-s + (0.123 + 0.992i)4-s + (−0.991 − 0.132i)5-s + (−0.345 + 0.938i)7-s + (0.564 − 0.825i)8-s + (0.654 + 0.755i)10-s + (0.683 + 0.730i)13-s + (0.879 − 0.475i)14-s + (−0.969 + 0.244i)16-s + (0.362 − 0.931i)17-s + (−0.774 + 0.633i)19-s + (0.00951 − 0.999i)20-s + (−0.786 + 0.618i)23-s + (0.964 + 0.263i)25-s + (−0.0285 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.544 + 0.838i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.544 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7655091197 + 0.4157090275i\)
\(L(\frac12)\) \(\approx\) \(0.7655091197 + 0.4157090275i\)
\(L(1)\) \(\approx\) \(0.6252136208 - 0.03228386109i\)
\(L(1)\) \(\approx\) \(0.6252136208 - 0.03228386109i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.749 - 0.662i)T \)
5 \( 1 + (-0.991 - 0.132i)T \)
7 \( 1 + (-0.345 + 0.938i)T \)
13 \( 1 + (0.683 + 0.730i)T \)
17 \( 1 + (0.362 - 0.931i)T \)
19 \( 1 + (-0.774 + 0.633i)T \)
23 \( 1 + (-0.786 + 0.618i)T \)
29 \( 1 + (0.710 + 0.703i)T \)
31 \( 1 + (0.820 - 0.572i)T \)
37 \( 1 + (0.516 - 0.856i)T \)
41 \( 1 + (0.595 + 0.803i)T \)
43 \( 1 + (-0.723 + 0.690i)T \)
47 \( 1 + (0.953 - 0.299i)T \)
53 \( 1 + (0.696 - 0.717i)T \)
59 \( 1 + (-0.398 - 0.917i)T \)
61 \( 1 + (0.948 + 0.318i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (-0.998 + 0.0570i)T \)
73 \( 1 + (-0.897 + 0.441i)T \)
79 \( 1 + (0.761 - 0.647i)T \)
83 \( 1 + (-0.999 - 0.0380i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.991 + 0.132i)T \)
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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

−20.75577808023595565625711445040, −20.10675240462467548061234650872, −19.41767674956185321740943495843, −18.914167623473731307507786065505, −17.90640548502439297488454896207, −17.173209401601719769702399301774, −16.476951688536463610464049891827, −15.65279151224852046563519252530, −15.20128793486298458782251643515, −14.22800227374887851228619035940, −13.41471343947321890887486978610, −12.40265812610225420350893533553, −11.35417911280953745635263084523, −10.400973113321647113090669205092, −10.254649881987649827145640022396, −8.73226418641155449094890099729, −8.23310773030776579786154560815, −7.46161139940513029933276790685, −6.62821610129622195654606528050, −5.92988236517811071117061920793, −4.56842548960955704338896405327, −3.89396905456933997270172865314, −2.65811915855414752969476370089, −1.10612891686000035528689550027, −0.38101608073849419392102473381, 0.748397057034617996084299314063, 1.93939326263835372040320038696, 2.94618083735096790629424557445, 3.77876475373686932476010470121, 4.63843646112461562029065160898, 6.01962009529362773919488908307, 6.97213708463076165624833600126, 7.95448342025903179313937860781, 8.562798167666721314803571438, 9.31849851698840947766411187861, 10.13972869638023651112227910773, 11.26695584872987736280739365712, 11.73597062730348490541830331231, 12.38504080232172177743500763125, 13.18744219848428714232182505433, 14.325969454126285054639830587131, 15.39939459658155329919823778293, 16.195741217761529143169387999679, 16.45600866805221926704140458112, 17.75952884397071656250373625291, 18.50156673856612041937448201454, 19.01733147017675029317819799067, 19.666063532526459697371840487264, 20.47560932984334513520447725704, 21.27759079872131280387565313723

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