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Properties

Label 1-33e2-1089.1028-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.570 + 0.821i$
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  + (−0.548 − 0.836i)2-s + (−0.398 + 0.917i)4-s + (0.532 + 0.846i)5-s + (0.797 + 0.603i)7-s + (0.985 − 0.170i)8-s + (0.415 − 0.909i)10-s + (0.861 − 0.508i)13-s + (0.0665 − 0.997i)14-s + (−0.683 − 0.730i)16-s + (−0.941 − 0.336i)17-s + (−0.0285 − 0.999i)19-s + (−0.988 + 0.151i)20-s + (0.327 + 0.945i)23-s + (−0.432 + 0.901i)25-s + (−0.897 − 0.441i)26-s + ⋯
L(s)  = 1  + (−0.548 − 0.836i)2-s + (−0.398 + 0.917i)4-s + (0.532 + 0.846i)5-s + (0.797 + 0.603i)7-s + (0.985 − 0.170i)8-s + (0.415 − 0.909i)10-s + (0.861 − 0.508i)13-s + (0.0665 − 0.997i)14-s + (−0.683 − 0.730i)16-s + (−0.941 − 0.336i)17-s + (−0.0285 − 0.999i)19-s + (−0.988 + 0.151i)20-s + (0.327 + 0.945i)23-s + (−0.432 + 0.901i)25-s + (−0.897 − 0.441i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.570 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3462278823 + 0.6619472468i\)
\(L(\frac12)\) \(\approx\) \(0.3462278823 + 0.6619472468i\)
\(L(1)\) \(\approx\) \(0.8351753445 - 0.04308599279i\)
\(L(1)\) \(\approx\) \(0.8351753445 - 0.04308599279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.548 - 0.836i)T \)
5 \( 1 + (0.532 + 0.846i)T \)
7 \( 1 + (0.797 + 0.603i)T \)
13 \( 1 + (0.861 - 0.508i)T \)
17 \( 1 + (-0.941 - 0.336i)T \)
19 \( 1 + (-0.0285 - 0.999i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (-0.997 - 0.0760i)T \)
31 \( 1 + (-0.948 - 0.318i)T \)
37 \( 1 + (-0.736 - 0.676i)T \)
41 \( 1 + (0.710 + 0.703i)T \)
43 \( 1 + (0.928 - 0.371i)T \)
47 \( 1 + (-0.161 + 0.986i)T \)
53 \( 1 + (-0.974 - 0.226i)T \)
59 \( 1 + (-0.964 - 0.263i)T \)
61 \( 1 + (0.449 + 0.893i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.610 - 0.791i)T \)
73 \( 1 + (0.516 + 0.856i)T \)
79 \( 1 + (0.272 - 0.962i)T \)
83 \( 1 + (-0.820 + 0.572i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.532 + 0.846i)T \)
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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

−20.62122727171938550685027775361, −20.42687559026154082967464421971, −19.27680191799730537417433582768, −18.38399155458195190462434504832, −17.77475950066970121929580030044, −16.947874770415426498110503560557, −16.541473701389502085899429479108, −15.693829928233900825737694335, −14.68781760663997153847310165682, −14.01756640732326847909259088236, −13.35940945599656466841719613165, −12.45202953097010224034846808570, −11.08672939592497949409452512076, −10.60065587346521098473036068280, −9.524332804174368108189759239688, −8.75395071168654574436849927036, −8.24017774815390953614327418814, −7.25038675061589754459183694210, −6.32082410135682013580523671203, −5.54287616899860564326633247850, −4.62958684538904381988421180685, −3.94928123806430498217639504723, −1.89030401285883804807652432930, −1.38754343493015479936908347596, −0.18017115020307631686317567625, 1.29561047364574247560050736153, 2.17377189838380690379270670688, 2.91431107514664243151496310079, 3.90899980737087839810532824376, 5.06018139919755892278536944899, 6.00092512476380547856755118712, 7.19000023980468913475507378391, 7.86808104672406891220301384999, 9.1167342467918093563460213804, 9.27169559172752795110755165596, 10.7232211401430502519455905535, 11.02074890873409658006808297157, 11.66063900542541195973084526046, 12.91021156638181426260090620611, 13.44167121244504014875583348536, 14.3565285541852937744801717560, 15.2871024677257602545239576340, 16.04400385993201251041941035991, 17.509929438757356557497953287530, 17.627265156235434085612095972525, 18.40206646665504083777610307534, 19.054652148675147503779232887005, 19.96460163147220528186799684962, 20.834327344757652473788575962525, 21.35671634010130652645517223710

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