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Properties

Label 1-33e2-1089.553-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.956 - 0.291i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.123 − 0.992i)2-s + (−0.969 − 0.244i)4-s + (0.964 − 0.263i)5-s + (−0.761 + 0.647i)7-s + (−0.362 + 0.931i)8-s + (−0.142 − 0.989i)10-s + (−0.0665 − 0.997i)13-s + (0.548 + 0.836i)14-s + (0.879 + 0.475i)16-s + (−0.736 + 0.676i)17-s + (0.198 + 0.980i)19-s + (−0.999 + 0.0190i)20-s + (0.235 + 0.971i)23-s + (0.861 − 0.508i)25-s + (−0.998 − 0.0570i)26-s + ⋯
L(s)  = 1  + (0.123 − 0.992i)2-s + (−0.969 − 0.244i)4-s + (0.964 − 0.263i)5-s + (−0.761 + 0.647i)7-s + (−0.362 + 0.931i)8-s + (−0.142 − 0.989i)10-s + (−0.0665 − 0.997i)13-s + (0.548 + 0.836i)14-s + (0.879 + 0.475i)16-s + (−0.736 + 0.676i)17-s + (0.198 + 0.980i)19-s + (−0.999 + 0.0190i)20-s + (0.235 + 0.971i)23-s + (0.861 − 0.508i)25-s + (−0.998 − 0.0570i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.956 - 0.291i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.956 - 0.291i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337955599 - 0.1990710399i\)
\(L(\frac12)\) \(\approx\) \(1.337955599 - 0.1990710399i\)
\(L(1)\) \(\approx\) \(1.001455361 - 0.3665811545i\)
\(L(1)\) \(\approx\) \(1.001455361 - 0.3665811545i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.123 - 0.992i)T \)
5 \( 1 + (0.964 - 0.263i)T \)
7 \( 1 + (-0.761 + 0.647i)T \)
13 \( 1 + (-0.0665 - 0.997i)T \)
17 \( 1 + (-0.736 + 0.676i)T \)
19 \( 1 + (0.198 + 0.980i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (0.00951 - 0.999i)T \)
31 \( 1 + (0.345 + 0.938i)T \)
37 \( 1 + (-0.466 + 0.884i)T \)
41 \( 1 + (-0.290 - 0.956i)T \)
43 \( 1 + (0.0475 + 0.998i)T \)
47 \( 1 + (0.820 + 0.572i)T \)
53 \( 1 + (-0.0285 + 0.999i)T \)
59 \( 1 + (-0.683 - 0.730i)T \)
61 \( 1 + (0.797 - 0.603i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (0.993 + 0.113i)T \)
73 \( 1 + (0.610 + 0.791i)T \)
79 \( 1 + (0.161 + 0.986i)T \)
83 \( 1 + (0.997 - 0.0760i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.964 + 0.263i)T \)
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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

−21.776850434469881914105092893227, −20.8522401872055280790982352630, −19.86995508991029609387333678792, −18.88513206802460531775186616533, −18.21517946353294197852450441147, −17.44557266412826628117808796350, −16.70164543550888547372409536190, −16.203359776811856239310270986060, −15.21753608251939299606742668920, −14.35225569019825913668374388110, −13.61953965971785915723941221501, −13.27690820639930332565116524935, −12.265806039488344227994572669115, −10.984944033331019799546403500251, −10.09732791885977955422559563515, −9.242551005902184360399272298602, −8.85143329896935476762466597408, −7.358136935993363349640985433554, −6.78593305811715397506433588155, −6.278634541578735120835624623525, −5.14083472852715433116121051022, −4.39579845314052888594346752983, −3.298059645645352238627363474512, −2.216441656114718551534956996229, −0.61069558217083578099472242955, 1.10853805622938266099285937171, 2.10930879332843469337954003296, 2.92072904098050572972352997149, 3.79921616128914514328096221377, 5.05722044982242207722342859693, 5.71366543509005019370276236321, 6.441746254441503299846409527100, 8.025660579075129261867603742556, 8.82293789941760214979473259079, 9.63662602521993209629753325059, 10.157509928684019706546693134484, 10.9803004500007641037536900787, 12.15608867137711083527791168276, 12.64527183350209887692418577576, 13.37384635900213616591845804797, 14.012739019456161753095622647526, 15.086472158450135605443113302387, 15.8191110386754279864972076084, 17.15300958792070123579641506814, 17.53846913875679708736744071248, 18.45671655666187647405710022378, 19.12784372564673159738321778922, 19.93321745808359259085053721487, 20.66912255810838760407753390053, 21.40991828672214738628661036045

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