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Properties

Label 2-1044-29.5-c1-0-8
Degree $2$
Conductor $1044$
Sign $0.503 + 0.864i$
Analytic cond. $8.33638$
Root an. cond. $2.88727$
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  + (1.81 + 2.27i)5-s + (−0.620 − 2.72i)7-s + (1.64 − 3.42i)11-s + (−5.43 − 2.61i)13-s − 6.61i·17-s + (2.85 + 0.652i)19-s + (1.28 − 1.61i)23-s + (−0.775 + 3.39i)25-s + (−2.03 − 4.98i)29-s + (0.575 − 0.459i)31-s + (5.06 − 6.35i)35-s + (2.51 + 5.21i)37-s + 1.73i·41-s + (6.44 + 5.14i)43-s + (−1.54 + 3.21i)47-s + ⋯
L(s)  = 1  + (0.812 + 1.01i)5-s + (−0.234 − 1.02i)7-s + (0.497 − 1.03i)11-s + (−1.50 − 0.725i)13-s − 1.60i·17-s + (0.655 + 0.149i)19-s + (0.268 − 0.337i)23-s + (−0.155 + 0.679i)25-s + (−0.376 − 0.926i)29-s + (0.103 − 0.0824i)31-s + (0.856 − 1.07i)35-s + (0.413 + 0.857i)37-s + 0.271i·41-s + (0.983 + 0.784i)43-s + (−0.225 + 0.469i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $0.503 + 0.864i$
Analytic conductor: \(8.33638\)
Root analytic conductor: \(2.88727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :1/2),\ 0.503 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.600324295\)
\(L(\frac12)\) \(\approx\) \(1.600324295\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 + (2.03 + 4.98i)T \)
good5 \( 1 + (-1.81 - 2.27i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.620 + 2.72i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (-1.64 + 3.42i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (5.43 + 2.61i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 6.61iT - 17T^{2} \)
19 \( 1 + (-2.85 - 0.652i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (-1.28 + 1.61i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (-0.575 + 0.459i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (-2.51 - 5.21i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 - 1.73iT - 41T^{2} \)
43 \( 1 + (-6.44 - 5.14i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (1.54 - 3.21i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.36 + 1.70i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + (13.3 - 3.04i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (4.97 - 2.39i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-12.2 - 5.88i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.225 - 0.179i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (5.36 + 11.1i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (-1.98 + 8.68i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-7.73 + 6.17i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (-0.521 - 0.119i)T + (87.3 + 42.0i)T^{2} \)
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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

−9.852959264147617459811580409506, −9.292990439455123849475086264280, −7.85211451422339720631201961743, −7.23828086772214147917842028600, −6.48690942163910251012228190924, −5.59652351189297863726778773008, −4.54068866784061931483285038555, −3.18920063646699134523038049631, −2.59425448960660183554858264737, −0.72882544630714577136279121666, 1.61688066020724189769790959752, 2.37513904921225290888360463279, 4.00100791023166274558792351184, 5.04589736213460976872563169059, 5.58700241764151048835673514963, 6.63563350084905221729160955676, 7.53678664076505218973050693280, 8.702716753311427125027949672654, 9.367771547846853027962201106068, 9.646427247729537256809538879355

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