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Properties

Label 1-33e2-1089.1037-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.600 - 0.799i$
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  + (−0.797 − 0.603i)2-s + (0.272 + 0.962i)4-s + (0.710 + 0.703i)5-s + (−0.179 − 0.983i)7-s + (0.362 − 0.931i)8-s + (−0.142 − 0.989i)10-s + (−0.830 + 0.556i)13-s + (−0.449 + 0.893i)14-s + (−0.851 + 0.524i)16-s + (0.736 − 0.676i)17-s + (0.198 + 0.980i)19-s + (−0.483 + 0.875i)20-s + (−0.723 + 0.690i)23-s + (0.00951 + 0.999i)25-s + (0.998 + 0.0570i)26-s + ⋯
L(s)  = 1  + (−0.797 − 0.603i)2-s + (0.272 + 0.962i)4-s + (0.710 + 0.703i)5-s + (−0.179 − 0.983i)7-s + (0.362 − 0.931i)8-s + (−0.142 − 0.989i)10-s + (−0.830 + 0.556i)13-s + (−0.449 + 0.893i)14-s + (−0.851 + 0.524i)16-s + (0.736 − 0.676i)17-s + (0.198 + 0.980i)19-s + (−0.483 + 0.875i)20-s + (−0.723 + 0.690i)23-s + (0.00951 + 0.999i)25-s + (0.998 + 0.0570i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.600 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3681361292 - 0.7371681936i\)
\(L(\frac12)\) \(\approx\) \(0.3681361292 - 0.7371681936i\)
\(L(1)\) \(\approx\) \(0.7161643832 - 0.1739562977i\)
\(L(1)\) \(\approx\) \(0.7161643832 - 0.1739562977i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.797 - 0.603i)T \)
5 \( 1 + (0.710 + 0.703i)T \)
7 \( 1 + (-0.179 - 0.983i)T \)
13 \( 1 + (-0.830 + 0.556i)T \)
17 \( 1 + (0.736 - 0.676i)T \)
19 \( 1 + (0.198 + 0.980i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (-0.861 - 0.508i)T \)
31 \( 1 + (0.640 - 0.768i)T \)
37 \( 1 + (-0.466 + 0.884i)T \)
41 \( 1 + (0.683 - 0.730i)T \)
43 \( 1 + (-0.888 - 0.458i)T \)
47 \( 1 + (0.905 - 0.424i)T \)
53 \( 1 + (0.0285 - 0.999i)T \)
59 \( 1 + (0.290 - 0.956i)T \)
61 \( 1 + (0.123 + 0.992i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.993 - 0.113i)T \)
73 \( 1 + (0.610 + 0.791i)T \)
79 \( 1 + (-0.935 - 0.353i)T \)
83 \( 1 + (0.432 - 0.901i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.710 + 0.703i)T \)
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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

−21.56548063616850984318969813438, −20.5744894363622790944192350194, −19.8086947201477383117146280715, −19.14195137337942022731301141906, −18.14968997463652653944285348662, −17.69900067247070182057528000340, −16.86278721088271273103800124924, −16.21153148349825997571691976238, −15.38344909001815371120127004564, −14.66718872359195917711394387363, −13.85849350008157154233470043606, −12.70717637133944087702403986172, −12.18765706520495916982692691649, −10.964636058811670521986660289667, −10.03003281248507753118649542236, −9.45005493076076461660260436310, −8.68954029302471296533929709154, −8.01931002294286933766620838883, −6.95236749503107110273941185265, −5.94594296469294904801858319323, −5.44872537868912288406681798852, −4.610375371038825947102793020684, −2.83000239737125985030804375162, −2.00241005215001102580746727637, −0.93869302313466801345123306307, 0.24742391629682657492987026366, 1.48478134691051853815983057898, 2.31142125631247260761601174398, 3.351309907834120689989868800202, 4.088744669576586147884156710172, 5.479153193202700335685506175895, 6.62417500528379932320385188123, 7.33685462757454338277948373695, 7.95072890973117564053654150665, 9.30355072938418779893847340553, 10.00347491917175081801338320076, 10.23646633457617512614549642086, 11.421828644476852942164368298534, 11.965082847295714250574734725208, 13.11677094462713206377387172001, 13.82821623539337396612488437902, 14.500790410167115056622729632367, 15.70904901122665043755534788109, 16.73572860697348847771243442244, 17.09354315910363141282053202397, 17.895322012351866558387176291259, 18.83588067195064912167344411401, 19.153681831451188102951305315841, 20.30873980443291410825333737995, 20.76168373609741222130989448843

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