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Properties

Label 1-33e2-1089.203-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.798 + 0.601i$
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  + (0.948 + 0.318i)2-s + (0.797 + 0.603i)4-s + (−0.380 + 0.924i)5-s + (0.640 − 0.768i)7-s + (0.564 + 0.825i)8-s + (−0.654 + 0.755i)10-s + (−0.290 − 0.956i)13-s + (0.851 − 0.524i)14-s + (0.272 + 0.962i)16-s + (0.362 + 0.931i)17-s + (0.774 + 0.633i)19-s + (−0.861 + 0.508i)20-s + (−0.928 + 0.371i)23-s + (−0.710 − 0.703i)25-s + (0.0285 − 0.999i)26-s + ⋯
L(s)  = 1  + (0.948 + 0.318i)2-s + (0.797 + 0.603i)4-s + (−0.380 + 0.924i)5-s + (0.640 − 0.768i)7-s + (0.564 + 0.825i)8-s + (−0.654 + 0.755i)10-s + (−0.290 − 0.956i)13-s + (0.851 − 0.524i)14-s + (0.272 + 0.962i)16-s + (0.362 + 0.931i)17-s + (0.774 + 0.633i)19-s + (−0.861 + 0.508i)20-s + (−0.928 + 0.371i)23-s + (−0.710 − 0.703i)25-s + (0.0285 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.798 + 0.601i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.798 + 0.601i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.023545313 + 3.058321456i\)
\(L(\frac12)\) \(\approx\) \(1.023545313 + 3.058321456i\)
\(L(1)\) \(\approx\) \(1.598896159 + 0.8127999502i\)
\(L(1)\) \(\approx\) \(1.598896159 + 0.8127999502i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.948 + 0.318i)T \)
5 \( 1 + (-0.380 + 0.924i)T \)
7 \( 1 + (0.640 - 0.768i)T \)
13 \( 1 + (-0.290 - 0.956i)T \)
17 \( 1 + (0.362 + 0.931i)T \)
19 \( 1 + (0.774 + 0.633i)T \)
23 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (-0.964 - 0.263i)T \)
31 \( 1 + (-0.905 + 0.424i)T \)
37 \( 1 + (0.516 + 0.856i)T \)
41 \( 1 + (0.398 + 0.917i)T \)
43 \( 1 + (0.235 - 0.971i)T \)
47 \( 1 + (0.217 + 0.976i)T \)
53 \( 1 + (-0.696 - 0.717i)T \)
59 \( 1 + (0.595 + 0.803i)T \)
61 \( 1 + (0.749 + 0.662i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (0.998 + 0.0570i)T \)
73 \( 1 + (0.897 + 0.441i)T \)
79 \( 1 + (-0.179 + 0.983i)T \)
83 \( 1 + (0.532 + 0.846i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (0.380 + 0.924i)T \)
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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

−20.856617382243904689686467294995, −20.43041640727145202151991566302, −19.61721513111249545056159588275, −18.763855165970548267579588134071, −17.97635273784794634885440981817, −16.69857997304585244588267784396, −16.13522723551505371381519920125, −15.43501238891135217389727664223, −14.49136146877269053695847685399, −13.9337056017061246606014300602, −12.91272473174547388361176223420, −12.21585160748341926439638432359, −11.60370351044064951024808189223, −11.05796970416233121393757868731, −9.56498919261119572019299188783, −9.132270194313028908506703108354, −7.85872097813695655380960532933, −7.153055277429031271493305875636, −5.86501044239058555531192412253, −5.19761496199641728550428813047, −4.52438961062470996028130431319, −3.64927926784578106927530688635, −2.39808267449134715039582120991, −1.65953281298530931819565954137, −0.43024302005965379004362460200, 1.36118385895158058616405259537, 2.51285417992999285046473002323, 3.59199829282265272129390212658, 3.99389807652422700814978908341, 5.24904622873708166099553312442, 5.966444686442094087190095391632, 6.981949055734456718854722532569, 7.81205236926439376049842199987, 8.05159577514843993418944354966, 9.89438509954186220792957025179, 10.64072542165535893740863161303, 11.309758057183121137007037750277, 12.10818430962676540081018341399, 12.99970335505861678870573944893, 13.87850734603216410101867755707, 14.574271387810109235194248937927, 15.01763584660396286636154428883, 15.920993581424711933076756493387, 16.77010980362376200689578155132, 17.59349045065859667146740118423, 18.29944249118546674668463226644, 19.48538002293931327196622518081, 20.14467639777647253994103442978, 20.8215520701621506942430428392, 21.771980337430828019224383414071

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