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Properties

Label 1-1053-1053.284-r0-0-0
Degree $1$
Conductor $1053$
Sign $-0.286 - 0.957i$
Analytic cond. $4.89011$
Root an. cond. $4.89011$
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  + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.448 − 0.893i)5-s + (0.727 + 0.686i)7-s + (−0.342 − 0.939i)8-s + (−0.766 + 0.642i)10-s + (0.549 − 0.835i)11-s + (−0.396 − 0.918i)14-s + (−0.0581 + 0.998i)16-s + (−0.939 − 0.342i)17-s + (0.984 + 0.173i)19-s + (0.957 − 0.286i)20-s + (−0.835 + 0.549i)22-s + (−0.686 − 0.727i)23-s + (−0.597 − 0.802i)25-s + ⋯
L(s)  = 1  + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.448 − 0.893i)5-s + (0.727 + 0.686i)7-s + (−0.342 − 0.939i)8-s + (−0.766 + 0.642i)10-s + (0.549 − 0.835i)11-s + (−0.396 − 0.918i)14-s + (−0.0581 + 0.998i)16-s + (−0.939 − 0.342i)17-s + (0.984 + 0.173i)19-s + (0.957 − 0.286i)20-s + (−0.835 + 0.549i)22-s + (−0.686 − 0.727i)23-s + (−0.597 − 0.802i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $-0.286 - 0.957i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ -0.286 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6239664141 - 0.8382822701i\)
\(L(\frac12)\) \(\approx\) \(0.6239664141 - 0.8382822701i\)
\(L(1)\) \(\approx\) \(0.7618109872 - 0.3283255330i\)
\(L(1)\) \(\approx\) \(0.7618109872 - 0.3283255330i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.918 - 0.396i)T \)
5 \( 1 + (0.448 - 0.893i)T \)
7 \( 1 + (0.727 + 0.686i)T \)
11 \( 1 + (0.549 - 0.835i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (-0.686 - 0.727i)T \)
29 \( 1 + (-0.597 - 0.802i)T \)
31 \( 1 + (0.727 - 0.686i)T \)
37 \( 1 + (-0.342 + 0.939i)T \)
41 \( 1 + (-0.802 - 0.597i)T \)
43 \( 1 + (0.0581 - 0.998i)T \)
47 \( 1 + (-0.727 - 0.686i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.549 - 0.835i)T \)
61 \( 1 + (-0.286 - 0.957i)T \)
67 \( 1 + (0.116 - 0.993i)T \)
71 \( 1 + (0.984 - 0.173i)T \)
73 \( 1 + (0.342 + 0.939i)T \)
79 \( 1 + (-0.993 + 0.116i)T \)
83 \( 1 + (0.116 + 0.993i)T \)
89 \( 1 + (0.984 + 0.173i)T \)
97 \( 1 + (0.549 - 0.835i)T \)
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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

−21.69665651835446347387634424131, −20.80604314768702578430644477927, −19.90288280715810497839927143233, −19.54466632860136232013465791504, −18.26512878494594047599526603591, −17.804681180140853259167791249322, −17.42714852739715152662400769556, −16.40729160726008484265910372014, −15.491396210216531653668620341678, −14.702882652211938854362812574490, −14.202229389708272879688805379103, −13.31168440487850851346728113243, −11.83352752091673919723222004453, −11.22085339973899109656761402635, −10.421948005311409546217630446820, −9.779746409666491190121723147319, −8.95711566077261632674215680180, −7.8487945876242960107727115025, −7.17027075460633836842164340856, −6.59324339914786788582509099102, −5.55863748229217542481595860397, −4.52495414012034377673891007531, −3.24859276679031577800369996083, −1.99713375720893115285263521078, −1.37531472911322107179882175201, 0.60507871063697973787762256277, 1.70405921205225265351717396658, 2.417052232540971202464893196689, 3.68714203912146678283894367224, 4.78171300950069128183234865711, 5.80051989128182472540491088467, 6.635332872412110557995661900864, 7.96508212882840883918427499596, 8.47008183244211043255381150980, 9.189421985631393910327068620394, 9.86379017664597302715644708628, 10.973190312623471189843628415885, 11.787157638940271370501172693988, 12.177571313558574339067152320240, 13.37231025993185692397641941041, 14.01198008469043188644056571597, 15.367829047807922248007343612196, 15.92181879936879732131579202775, 16.97302209694439889515991545721, 17.25671868670348710904153931563, 18.37757028816112359513820739688, 18.68442831703071420580101996686, 19.918526689992005318157029798951, 20.39893287414427874924059031720, 21.110530774104284448237473697603

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