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Properties

Label 2-1044-29.6-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: $\chi_{1044} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :1/2),\ -0.503 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2083947788\)
\(L(\frac12)\) \(\approx\) \(0.2083947788\)
\(L(\frac{3}{2})\) 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

−9.532979979151305115845070979975, −8.964967208758101166938742534648, −7.71141598059294846926799703873, −7.32635720987838856504454209134, −6.28608464283657676544116223402, −5.36426260835046636720276286991, −4.34082032865484201063717955097, −2.89678483639040356861892826710, −2.63444447840162002729532966358, −0.094958501271761957636730747647, 1.40004525557838708907610830944, 3.04101778132720293545384924309, 4.28533927973303094487905186528, 4.70828958770928080710713365335, 5.82008985334681965338557679334, 7.22494129212732565099598692062, 7.63106472253997117503012497873, 8.364782213772673390025618445162, 9.485655796693140470133320876542, 10.19893046379364370576336115876

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