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Properties

Label 1-33e2-1089.562-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  + (0.928 + 0.371i)2-s + (0.723 + 0.690i)4-s + (−0.995 + 0.0950i)5-s + (−0.888 + 0.458i)7-s + (0.415 + 0.909i)8-s + (−0.959 − 0.281i)10-s + (0.723 + 0.690i)13-s + (−0.995 + 0.0950i)14-s + (0.0475 + 0.998i)16-s + (−0.654 − 0.755i)17-s + (−0.654 + 0.755i)19-s + (−0.786 − 0.618i)20-s + (−0.888 − 0.458i)23-s + (0.981 − 0.189i)25-s + (0.415 + 0.909i)26-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (0.723 + 0.690i)4-s + (−0.995 + 0.0950i)5-s + (−0.888 + 0.458i)7-s + (0.415 + 0.909i)8-s + (−0.959 − 0.281i)10-s + (0.723 + 0.690i)13-s + (−0.995 + 0.0950i)14-s + (0.0475 + 0.998i)16-s + (−0.654 − 0.755i)17-s + (−0.654 + 0.755i)19-s + (−0.786 − 0.618i)20-s + (−0.888 − 0.458i)23-s + (0.981 − 0.189i)25-s + (0.415 + 0.909i)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: $\chi_{1089} (562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.941 - 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1243082704 + 0.7170778087i\)
\(L(\frac12)\) \(\approx\) \(-0.1243082704 + 0.7170778087i\)
\(L(1)\) \(\approx\) \(0.9952748223 + 0.5166451460i\)
\(L(1)\) \(\approx\) \(0.9952748223 + 0.5166451460i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.928 + 0.371i)T \)
5 \( 1 + (-0.995 + 0.0950i)T \)
7 \( 1 + (-0.888 + 0.458i)T \)
13 \( 1 + (0.723 + 0.690i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (-0.327 - 0.945i)T \)
31 \( 1 + (0.235 + 0.971i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (-0.786 + 0.618i)T \)
43 \( 1 + (-0.995 - 0.0950i)T \)
47 \( 1 + (-0.786 - 0.618i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.928 - 0.371i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (0.580 + 0.814i)T \)
83 \( 1 + (-0.888 + 0.458i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.995 - 0.0950i)T \)
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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

−20.92409210827027893735643859297, −20.16911327733670228637116797541, −19.64094923911436556794152117099, −19.10826953540843730778147761710, −18.05243789716927167606095185494, −16.82610509240864047707093723554, −16.0854491810186045965505672121, −15.382029094860759876113632877746, −14.93414210442854425242051545493, −13.61488966127727866062230501744, −13.161008585913371544855870399086, −12.42715270399261747773355168185, −11.583463284121228105267690005786, −10.76673735210962344774821041285, −10.21449444595583672306976449923, −8.97929464430567050421598316337, −7.99552229589537894767009502777, −6.958620771164669929604409117095, −6.354852487075807189745037400107, −5.313373020229963320699714705287, −4.213580491428944090162932461709, −3.68162396348638235251250963213, −2.90403424494930735600078790031, −1.57548493851124733061536759624, −0.20324270052508011837743557467, 1.91545065982141801264407358853, 2.99080715137166014957920143781, 3.7880602162700392823675676875, 4.44611084159772640842831165779, 5.56479636412505563119499658779, 6.570846851289920649278341167950, 6.93419703227738196155328067571, 8.20590416646231246656980347791, 8.69506841940100853535715289753, 10.027258071874349895715343540495, 11.06970996255147530787564728385, 11.862407855419328935915595923219, 12.33099923393568867513554798782, 13.28914598425116574545754988452, 13.97904897760451277095220144220, 14.97581042722641520034462222548, 15.5889262337027378582949801923, 16.24615952093540531599280713813, 16.70794121780458473451271190679, 18.07495867300887423857613045964, 18.8723075130945134643210696429, 19.66357409443071658967398047824, 20.37479728824373526237910071697, 21.234446626500711639972324567041, 22.02924001928696261603010406125

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