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Properties

Label 1-33e2-1089.232-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.945 - 0.325i$
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  + (0.580 − 0.814i)2-s + (−0.327 − 0.945i)4-s + (0.235 − 0.971i)5-s + (0.928 − 0.371i)7-s + (−0.959 − 0.281i)8-s + (−0.654 − 0.755i)10-s + (−0.327 − 0.945i)13-s + (0.235 − 0.971i)14-s + (−0.786 + 0.618i)16-s + (0.841 − 0.540i)17-s + (0.841 + 0.540i)19-s + (−0.995 + 0.0950i)20-s + (0.928 + 0.371i)23-s + (−0.888 − 0.458i)25-s + (−0.959 − 0.281i)26-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (−0.327 − 0.945i)4-s + (0.235 − 0.971i)5-s + (0.928 − 0.371i)7-s + (−0.959 − 0.281i)8-s + (−0.654 − 0.755i)10-s + (−0.327 − 0.945i)13-s + (0.235 − 0.971i)14-s + (−0.786 + 0.618i)16-s + (0.841 − 0.540i)17-s + (0.841 + 0.540i)19-s + (−0.995 + 0.0950i)20-s + (0.928 + 0.371i)23-s + (−0.888 − 0.458i)25-s + (−0.959 − 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.945 - 0.325i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.945 - 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3742460642 - 2.235415887i\)
\(L(\frac12)\) \(\approx\) \(0.3742460642 - 2.235415887i\)
\(L(1)\) \(\approx\) \(1.054886603 - 1.154999251i\)
\(L(1)\) \(\approx\) \(1.054886603 - 1.154999251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.580 - 0.814i)T \)
5 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (0.928 - 0.371i)T \)
13 \( 1 + (-0.327 - 0.945i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (0.0475 - 0.998i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (-0.995 - 0.0950i)T \)
43 \( 1 + (0.235 + 0.971i)T \)
47 \( 1 + (-0.995 + 0.0950i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (-0.995 + 0.0950i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.235 + 0.971i)T \)
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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

−21.85933979195280147179495537026, −21.29581167471763260262203456818, −20.58879034269576768326449624517, −19.1745886379300612086525685852, −18.5497455791854534603511213345, −17.75360249710625682594189895301, −17.14179424186265560933947940256, −16.272832198904274498929887560646, −15.23743515230677653411251218205, −14.77883281591745255819865282047, −14.08944370530883039217105526949, −13.50790670990563919194106193635, −12.24885008883893058347900338474, −11.68895480229822724423059678889, −10.81336040746903300917673548192, −9.706604604274339509348571631433, −8.74432542102593378856430652336, −7.932650063112962148575581050096, −7.01366927536289208804613477305, −6.509236628923964636504736473984, −5.329087157807873517740612527754, −4.82560064282094352949550326745, −3.576191151811907160433055229580, −2.80383242168347268012043344083, −1.663964730666351360087738951775, 0.835998908840943470874956509627, 1.471654832253835692705544721143, 2.676805681536704662882112029937, 3.67121236960528408714439597015, 4.7607034957669693241656555173, 5.190391629376527409615018296353, 5.99972539583682496120702832052, 7.507607264897552069964580373492, 8.23660541936383426532028418216, 9.32980231557055894322988202967, 9.988079932398929890991411975718, 10.81690113902317086321467546346, 11.86449954909760860736204328268, 12.200097020425305266587681504581, 13.31673279030360375787169547181, 13.74718225495921932581058426220, 14.66448699138158895413990250955, 15.41878852074212191563649753930, 16.436623328877474433621018649278, 17.39958714784426577396230843296, 17.94505097082139353463505782772, 18.96025142460913048587604582627, 19.812317667831915898657911002395, 20.5444500211671521089097269962, 20.96757524796476457273120325727

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