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Properties

Label 1-1083-1083.143-r0-0-0
Degree $1$
Conductor $1083$
Sign $-0.378 - 0.925i$
Analytic cond. $5.02943$
Root an. cond. $5.02943$
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  + (0.0642 − 0.997i)2-s + (−0.991 − 0.128i)4-s + (0.703 − 0.710i)5-s + (0.975 + 0.218i)7-s + (−0.191 + 0.981i)8-s + (−0.663 − 0.748i)10-s + (0.962 + 0.272i)11-s + (−0.741 − 0.670i)13-s + (0.280 − 0.959i)14-s + (0.967 + 0.254i)16-s + (−0.384 + 0.922i)17-s + (−0.789 + 0.614i)20-s + (0.333 − 0.942i)22-s + (−0.999 + 0.0367i)23-s + (−0.00918 − 0.999i)25-s + (−0.716 + 0.697i)26-s + ⋯
L(s)  = 1  + (0.0642 − 0.997i)2-s + (−0.991 − 0.128i)4-s + (0.703 − 0.710i)5-s + (0.975 + 0.218i)7-s + (−0.191 + 0.981i)8-s + (−0.663 − 0.748i)10-s + (0.962 + 0.272i)11-s + (−0.741 − 0.670i)13-s + (0.280 − 0.959i)14-s + (0.967 + 0.254i)16-s + (−0.384 + 0.922i)17-s + (−0.789 + 0.614i)20-s + (0.333 − 0.942i)22-s + (−0.999 + 0.0367i)23-s + (−0.00918 − 0.999i)25-s + (−0.716 + 0.697i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $-0.378 - 0.925i$
Analytic conductor: \(5.02943\)
Root analytic conductor: \(5.02943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1083,\ (0:\ ),\ -0.378 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001222226 - 1.490427214i\)
\(L(\frac12)\) \(\approx\) \(1.001222226 - 1.490427214i\)
\(L(1)\) \(\approx\) \(1.023391443 - 0.7428535553i\)
\(L(1)\) \(\approx\) \(1.023391443 - 0.7428535553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0642 - 0.997i)T \)
5 \( 1 + (0.703 - 0.710i)T \)
7 \( 1 + (0.975 + 0.218i)T \)
11 \( 1 + (0.962 + 0.272i)T \)
13 \( 1 + (-0.741 - 0.670i)T \)
17 \( 1 + (-0.384 + 0.922i)T \)
23 \( 1 + (-0.999 + 0.0367i)T \)
29 \( 1 + (0.888 - 0.459i)T \)
31 \( 1 + (0.592 - 0.805i)T \)
37 \( 1 + (-0.245 - 0.969i)T \)
41 \( 1 + (0.919 - 0.393i)T \)
43 \( 1 + (-0.979 - 0.200i)T \)
47 \( 1 + (-0.100 + 0.994i)T \)
53 \( 1 + (0.811 + 0.584i)T \)
59 \( 1 + (-0.800 - 0.599i)T \)
61 \( 1 + (0.933 - 0.359i)T \)
67 \( 1 + (0.999 + 0.0183i)T \)
71 \( 1 + (0.933 + 0.359i)T \)
73 \( 1 + (0.606 + 0.794i)T \)
79 \( 1 + (-0.315 - 0.948i)T \)
83 \( 1 + (0.821 - 0.569i)T \)
89 \( 1 + (0.606 - 0.794i)T \)
97 \( 1 + (0.999 - 0.0183i)T \)
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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

−21.717353544599713685401569824852, −21.36955997718104016524450461910, −20.058207478190413459207426864418, −19.187124243108145466814550716683, −18.15353912491975063898813951657, −17.855375144759074255060153166173, −16.99965900573253673609976783004, −16.39313961430063731016007270938, −15.27486247097669726565915220122, −14.53404948547576052346976099758, −14.0052182220781661995725554388, −13.59861249495801596544967541410, −12.163597913498028633633977684759, −11.49262867153833760114092730868, −10.3229547540308878287660039817, −9.59836531805884580319018493822, −8.75922371732905277117749140557, −7.890828546722116187932476562017, −6.79705018332356549787172705798, −6.58969701628786406592724701777, −5.31793848131243716948402107299, −4.66878355106612452058777894127, −3.65955181285667206255739473726, −2.38844339953674410161519771551, −1.20066748155758766851703648534, 0.85576969407083745147634730806, 1.85937090556758704917884338438, 2.41879714333535981633835537394, 3.94042643444752283603573009927, 4.57745722982305970703729782520, 5.43369117344852933914893596078, 6.25041858519734796247000154191, 7.86129362866339989616246373526, 8.491273130847170161456640438686, 9.35652156704479878917215220444, 10.03642764585823676195366058818, 10.86751838743974580748287447614, 11.8885238471206035285657277633, 12.35031372990871713599940613487, 13.17169792476735374819934086496, 14.128096349458366918342876878912, 14.5730683003959966521602617241, 15.620988810328230837904282129441, 17.08237534470006964711760408285, 17.41286271624946682211589396751, 17.9566193068654081621375568310, 19.05426094098055497293851574914, 19.96060386786759814532534376755, 20.2995563705021409214896368589, 21.322342037665295975182991612329

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