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Properties

Label 1-33e2-1089.178-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.874 - 0.484i$
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  + (0.179 + 0.983i)2-s + (−0.935 + 0.353i)4-s + (−0.683 + 0.730i)5-s + (−0.953 − 0.299i)7-s + (−0.516 − 0.856i)8-s + (−0.841 − 0.540i)10-s + (−0.548 + 0.836i)13-s + (0.123 − 0.992i)14-s + (0.749 − 0.662i)16-s + (0.466 + 0.884i)17-s + (0.985 + 0.170i)19-s + (0.380 − 0.924i)20-s + (0.580 + 0.814i)23-s + (−0.0665 − 0.997i)25-s + (−0.921 − 0.389i)26-s + ⋯
L(s)  = 1  + (0.179 + 0.983i)2-s + (−0.935 + 0.353i)4-s + (−0.683 + 0.730i)5-s + (−0.953 − 0.299i)7-s + (−0.516 − 0.856i)8-s + (−0.841 − 0.540i)10-s + (−0.548 + 0.836i)13-s + (0.123 − 0.992i)14-s + (0.749 − 0.662i)16-s + (0.466 + 0.884i)17-s + (0.985 + 0.170i)19-s + (0.380 − 0.924i)20-s + (0.580 + 0.814i)23-s + (−0.0665 − 0.997i)25-s + (−0.921 − 0.389i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.874 - 0.484i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.874 - 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3264172948 + 1.263289539i\)
\(L(\frac12)\) \(\approx\) \(-0.3264172948 + 1.263289539i\)
\(L(1)\) \(\approx\) \(0.5919819171 + 0.6349449605i\)
\(L(1)\) \(\approx\) \(0.5919819171 + 0.6349449605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.179 + 0.983i)T \)
5 \( 1 + (-0.683 + 0.730i)T \)
7 \( 1 + (-0.953 - 0.299i)T \)
13 \( 1 + (-0.548 + 0.836i)T \)
17 \( 1 + (0.466 + 0.884i)T \)
19 \( 1 + (0.985 + 0.170i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (0.830 + 0.556i)T \)
31 \( 1 + (0.988 + 0.151i)T \)
37 \( 1 + (-0.254 + 0.967i)T \)
41 \( 1 + (0.851 + 0.524i)T \)
43 \( 1 + (-0.981 - 0.189i)T \)
47 \( 1 + (0.997 + 0.0760i)T \)
53 \( 1 + (0.198 - 0.980i)T \)
59 \( 1 + (0.879 + 0.475i)T \)
61 \( 1 + (0.761 + 0.647i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (-0.993 - 0.113i)T \)
79 \( 1 + (-0.820 + 0.572i)T \)
83 \( 1 + (-0.00951 - 0.999i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.683 - 0.730i)T \)
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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

−20.57605237756123419666823105397, −20.1211588803217747871644705065, −19.325047809840049656628128096359, −18.818346066495254406531438002459, −17.83854234440591919640907406123, −16.976749960477469682479063524044, −15.99349449152748475357189949967, −15.41958800422984371887533233869, −14.335451606126510464706181966183, −13.44324160840505591143573268213, −12.6784484454862841806687181962, −12.15656980434132044451407092195, −11.508867760115256136236297755977, −10.37107964607121975968786056453, −9.67248645086305986150311234481, −8.96058203455410898351344726743, −8.08802918021975530242984364189, −7.09001463471835261053915505524, −5.71248083228578052514310693411, −5.04693149462086954115972876042, −4.13359418968827245774790668580, −3.12020494343819070504116705296, −2.552550932660766320794880692516, −0.87988795651916705740774296349, −0.40599982696692159382631154204, 0.97817131445754953861690592701, 2.87756420771214459769329558872, 3.567047023205253819465799142818, 4.38844634665685483802538471380, 5.47649684546728234967195987299, 6.53757096005037492007168513208, 6.97873790679492861640525713472, 7.772294504648771907605675988593, 8.67297063704568244379179749132, 9.7420489718304927285864529318, 10.26153833828634858026522808063, 11.62239432909998370039584227773, 12.27538640972135215319732733816, 13.25086498130979760796194254944, 14.027079346239533545131012575681, 14.710777596206234053255519400076, 15.52603660997032640283060837354, 16.13277569517457879547187139857, 16.858431277915813797481763574579, 17.66001951925728428702960532559, 18.66071751549787522097497274864, 19.213512791660923996317835168877, 19.830072251592338369083332647747, 21.21696834188524000668688755025, 21.99015316722715940138383167054

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