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Properties

Label 2-1083-57.56-c1-0-29
Degree $2$
Conductor $1083$
Sign $0.519 - 0.854i$
Analytic cond. $8.64779$
Root an. cond. $2.94071$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.517·2-s + (1.41 − i)3-s − 1.73·4-s + 1.41i·5-s + (0.732 − 0.517i)6-s − 0.267·7-s − 1.93·8-s + (1.00 − 2.82i)9-s + 0.732i·10-s + 5.27i·11-s + (−2.44 + 1.73i)12-s + 0.267i·13-s − 0.138·14-s + (1.41 + 2.00i)15-s + 2.46·16-s + 4.89i·17-s + ⋯
L(s)  = 1  + 0.366·2-s + (0.816 − 0.577i)3-s − 0.866·4-s + 0.632i·5-s + (0.298 − 0.211i)6-s − 0.101·7-s − 0.683·8-s + (0.333 − 0.942i)9-s + 0.231i·10-s + 1.59i·11-s + (−0.707 + 0.499i)12-s + 0.0743i·13-s − 0.0370·14-s + (0.365 + 0.516i)15-s + 0.616·16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $0.519 - 0.854i$
Analytic conductor: \(8.64779\)
Root analytic conductor: \(2.94071\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1083,\ (\ :1/2),\ 0.519 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.782894542\)
\(L(\frac12)\) \(\approx\) \(1.782894542\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.41 + i)T \)
19 \( 1 \)
good2 \( 1 - 0.517T + 2T^{2} \)
5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 + 0.267T + 7T^{2} \)
11 \( 1 - 5.27iT - 11T^{2} \)
13 \( 1 - 0.267iT - 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
23 \( 1 - 5.27iT - 23T^{2} \)
29 \( 1 + 2.07T + 29T^{2} \)
31 \( 1 + 2.46iT - 31T^{2} \)
37 \( 1 - 7.73iT - 37T^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 - 5.73T + 43T^{2} \)
47 \( 1 - 0.757iT - 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - iT - 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + 2.07iT - 83T^{2} \)
89 \( 1 + 7.34T + 89T^{2} \)
97 \( 1 + 0.535iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.717757388875033261902443082455, −9.319521438544965038777615756401, −8.275802544687072034364535391841, −7.55900584929255513726857843793, −6.74685028153861883938429198881, −5.82945682377104054036688875042, −4.56026670473156527395521274181, −3.79545773859170585622150382567, −2.80704583610088802629407795899, −1.57674229206542741181515339748, 0.68598035566935325685122181619, 2.71893372170014357541654637429, 3.53775319364651461743774361407, 4.48409450796723722325060316351, 5.18040371282205162837777939763, 6.05784966149128916899858693516, 7.50763711682224552226066169477, 8.416037566939934956899281694558, 8.971278582480331228677326305983, 9.380587170662615319961206562598

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