Properties

Label 1-33e2-1089.277-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.574 - 0.818i$
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  + (−0.820 + 0.572i)2-s + (0.345 − 0.938i)4-s + (0.449 − 0.893i)5-s + (0.432 − 0.901i)7-s + (0.254 + 0.967i)8-s + (0.142 + 0.989i)10-s + (−0.272 − 0.962i)13-s + (0.161 + 0.986i)14-s + (−0.761 − 0.647i)16-s + (0.870 + 0.491i)17-s + (0.736 + 0.676i)19-s + (−0.683 − 0.730i)20-s + (0.723 − 0.690i)23-s + (−0.595 − 0.803i)25-s + (0.774 + 0.633i)26-s + ⋯
L(s)  = 1  + (−0.820 + 0.572i)2-s + (0.345 − 0.938i)4-s + (0.449 − 0.893i)5-s + (0.432 − 0.901i)7-s + (0.254 + 0.967i)8-s + (0.142 + 0.989i)10-s + (−0.272 − 0.962i)13-s + (0.161 + 0.986i)14-s + (−0.761 − 0.647i)16-s + (0.870 + 0.491i)17-s + (0.736 + 0.676i)19-s + (−0.683 − 0.730i)20-s + (0.723 − 0.690i)23-s + (−0.595 − 0.803i)25-s + (0.774 + 0.633i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.574 - 0.818i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.574 - 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.749094891 - 0.9090786771i\)
\(L(\frac12)\) \(\approx\) \(1.749094891 - 0.9090786771i\)
\(L(1)\) \(\approx\) \(0.9640204486 - 0.1347869069i\)
\(L(1)\) \(\approx\) \(0.9640204486 - 0.1347869069i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.820 + 0.572i)T \)
5 \( 1 + (0.449 - 0.893i)T \)
7 \( 1 + (0.432 - 0.901i)T \)
13 \( 1 + (-0.272 - 0.962i)T \)
17 \( 1 + (0.870 + 0.491i)T \)
19 \( 1 + (0.736 + 0.676i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (0.398 + 0.917i)T \)
31 \( 1 + (-0.532 - 0.846i)T \)
37 \( 1 + (0.897 - 0.441i)T \)
41 \( 1 + (-0.123 + 0.992i)T \)
43 \( 1 + (0.888 + 0.458i)T \)
47 \( 1 + (0.483 - 0.875i)T \)
53 \( 1 + (0.941 + 0.336i)T \)
59 \( 1 + (0.797 - 0.603i)T \)
61 \( 1 + (0.905 - 0.424i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (0.0285 + 0.999i)T \)
79 \( 1 + (0.625 - 0.780i)T \)
83 \( 1 + (0.991 + 0.132i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.449 + 0.893i)T \)
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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

−21.2241565419251016140162389512, −20.79319652585477307067216501420, −19.45347960182559883246180384975, −19.0471298901230788325354851858, −18.26886595497636833478980695320, −17.74892323060708624095677271425, −16.951422428513442918776210725002, −15.97362850340324638460091175281, −15.21031593362992574892493673678, −14.256072740206859909152480493, −13.522770342091699625199538769729, −12.35578962925499576809054422515, −11.61965445874456857284898837738, −11.14905520353005696655613945931, −10.09121346699515357061646275271, −9.400869021894498862538309712347, −8.79798929298866753387963206391, −7.57564779304039007086166463418, −7.0659068596006502245021176153, −5.979648282321904440089210411367, −4.9542691880990679648026082797, −3.61114667130349871510883024435, −2.69397758998215074022285427914, −2.08443146976022798487757397306, −0.94827298613890240215161959101, 0.74531462607984137349877312067, 1.11524689045059250588336021660, 2.36317754340689947861410842228, 3.83851564972848261238683599390, 5.05215096232419339771435839340, 5.52665885226974823211240904033, 6.58328353906330605980634717121, 7.70053835667298813458133889677, 8.05562982954245058928910816511, 9.05390637104508258277610236286, 9.94524130329316284456894179952, 10.43459136969421375110274418949, 11.40642664132832402447170667994, 12.53667535622021668819570050230, 13.29553271905118465976018581974, 14.35898465463327110340812721800, 14.76905313458897924866750701115, 15.99947501614690892636401647181, 16.630022230059856606931410654978, 17.131574847483569638181648193583, 17.874288989930937520190729405352, 18.58312754471423377785010001621, 19.673015986924326613580340437012, 20.33471128279526198752340202553, 20.71277313549094261369820761673

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