Properties

Label 1-33e2-1089.38-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.819 + 0.572i$
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  + (0.905 − 0.424i)2-s + (0.640 − 0.768i)4-s + (−0.548 + 0.836i)5-s + (0.997 + 0.0760i)7-s + (0.254 − 0.967i)8-s + (−0.142 + 0.989i)10-s + (−0.969 + 0.244i)13-s + (0.935 − 0.353i)14-s + (−0.179 − 0.983i)16-s + (0.870 − 0.491i)17-s + (−0.736 + 0.676i)19-s + (0.290 + 0.956i)20-s + (−0.235 + 0.971i)23-s + (−0.398 − 0.917i)25-s + (−0.774 + 0.633i)26-s + ⋯
L(s)  = 1  + (0.905 − 0.424i)2-s + (0.640 − 0.768i)4-s + (−0.548 + 0.836i)5-s + (0.997 + 0.0760i)7-s + (0.254 − 0.967i)8-s + (−0.142 + 0.989i)10-s + (−0.969 + 0.244i)13-s + (0.935 − 0.353i)14-s + (−0.179 − 0.983i)16-s + (0.870 − 0.491i)17-s + (−0.736 + 0.676i)19-s + (0.290 + 0.956i)20-s + (−0.235 + 0.971i)23-s + (−0.398 − 0.917i)25-s + (−0.774 + 0.633i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.819 + 0.572i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.819 + 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.421495852 + 1.076352358i\)
\(L(\frac12)\) \(\approx\) \(3.421495852 + 1.076352358i\)
\(L(1)\) \(\approx\) \(1.800561348 - 0.09275360057i\)
\(L(1)\) \(\approx\) \(1.800561348 - 0.09275360057i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.905 - 0.424i)T \)
5 \( 1 + (-0.548 + 0.836i)T \)
7 \( 1 + (0.997 + 0.0760i)T \)
13 \( 1 + (-0.969 + 0.244i)T \)
17 \( 1 + (0.870 - 0.491i)T \)
19 \( 1 + (-0.736 + 0.676i)T \)
23 \( 1 + (-0.235 + 0.971i)T \)
29 \( 1 + (0.595 + 0.803i)T \)
31 \( 1 + (0.999 + 0.0380i)T \)
37 \( 1 + (0.897 + 0.441i)T \)
41 \( 1 + (-0.797 + 0.603i)T \)
43 \( 1 + (0.0475 - 0.998i)T \)
47 \( 1 + (0.999 + 0.0190i)T \)
53 \( 1 + (-0.941 + 0.336i)T \)
59 \( 1 + (-0.123 + 0.992i)T \)
61 \( 1 + (0.820 - 0.572i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (-0.198 + 0.980i)T \)
73 \( 1 + (-0.0285 + 0.999i)T \)
79 \( 1 + (0.988 - 0.151i)T \)
83 \( 1 + (-0.380 + 0.924i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (0.548 + 0.836i)T \)
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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

−21.14588190165489061780991017260, −20.64816821053645276627977549325, −19.82112280436439038829947384303, −19.06368586242582129214516338219, −17.635400699774975933301373626028, −17.15875924338952017166318032608, −16.46302941025097809050149654631, −15.52727840119386783702030724478, −14.88091408625833277991549347460, −14.253745578374439378013449023576, −13.26766039597311318690060250671, −12.45358345519982541316820765188, −11.95414184633960700428114693904, −11.13836335598148566597846748040, −10.133588260346340210211939876682, −8.760345784773602956841051011, −8.056752302380218774994164729232, −7.552072656556399752660041782, −6.40516133223652393467689009980, −5.40634712747195117434767686546, −4.60164180125866525776435885125, −4.22243700228030713683887074638, −2.8893471048241764775026101065, −1.88926656160917734193163297611, −0.55230916355811284815932792168, 1.07640551076774752493007595098, 2.18076012546018995958641942593, 2.98638352753057465731973040627, 3.96645695670928936100047936804, 4.76504943379959733211870445543, 5.603146310253468776806937185319, 6.6591219131224506462232675647, 7.45044630870671621671499447854, 8.196585560195775529605288817321, 9.67802633922340674151931477961, 10.392381025449822382122450157851, 11.17270294365323415740415912928, 11.95925268101986152724863393402, 12.3012345658701083708031620773, 13.71259798311932799262125708178, 14.26384120744289932790925384190, 14.89826033259756852480438348149, 15.47053301138074142393550831436, 16.50787147791199724029475946411, 17.47902298344034197295291113014, 18.5495866052347639960367175383, 19.026145207110348926242928488279, 19.893265251163078806565741066055, 20.61427825166842082178155048806, 21.570635759263594281380442446286

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