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Properties

Label 2-24e2-9.5-c2-0-30
Degree $2$
Conductor $576$
Sign $0.642 + 0.766i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + (−6.39 − 3.69i)5-s + (3.39 + 5.88i)7-s + 9·9-s + (−5.29 + 3.05i)11-s + (8.39 − 14.5i)13-s + (−19.1 − 11.0i)15-s − 25.1i·17-s + 17.5·19-s + (10.1 + 17.6i)21-s + (12.3 + 7.15i)23-s + (14.7 + 25.6i)25-s + 27·27-s + (−16.1 + 9.35i)29-s + (23.3 − 40.5i)31-s + ⋯
L(s)  = 1  + 3-s + (−1.27 − 0.738i)5-s + (0.485 + 0.841i)7-s + 9-s + (−0.481 + 0.278i)11-s + (0.646 − 1.11i)13-s + (−1.27 − 0.738i)15-s − 1.48i·17-s + 0.926·19-s + (0.485 + 0.841i)21-s + (0.539 + 0.311i)23-s + (0.591 + 1.02i)25-s + 27-s + (−0.558 + 0.322i)29-s + (0.754 − 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.147654323\)
\(L(\frac12)\) \(\approx\) \(2.147654323\)
\(L(2)\) 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 \( 1 - 3T \)
good5 \( 1 + (6.39 + 3.69i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.39 - 5.88i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.29 - 3.05i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.39 + 14.5i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 25.1iT - 289T^{2} \)
19 \( 1 - 17.5T + 361T^{2} \)
23 \( 1 + (-12.3 - 7.15i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (16.1 - 9.35i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-23.3 + 40.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 49.5T + 1.36e3T^{2} \)
41 \( 1 + (34.5 + 19.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.0 + 38.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28.8 + 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 10.1iT - 2.80e3T^{2} \)
59 \( 1 + (-14.2 - 8.25i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-10.6 - 18.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (43.4 - 75.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 30.2iT - 5.04e3T^{2} \)
73 \( 1 + 48.7T + 5.32e3T^{2} \)
79 \( 1 + (55.7 + 96.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-85.0 + 49.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 75.5iT - 7.92e3T^{2} \)
97 \( 1 + (-70.2 - 121. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.27126109448348416178407902061, −9.215501117587402707515103747635, −8.591622121192660926524283273049, −7.78933348591376497563014955290, −7.34482980624141117809351449002, −5.49989184037788732915779209031, −4.69977025696506042830821918527, −3.55177177308958986228403185427, −2.56718656808557055455405157069, −0.837743817370796911723038823222, 1.37819337430813906522954436498, 3.02964242668641056997276204255, 3.85027344442195721233138870437, 4.56866169049821793774190876385, 6.45222081320861973693901882149, 7.29456403412697544876071671020, 7.996976705590466548948787683897, 8.575978625421009749878157509471, 9.798358772912798521156037066173, 10.79275080393216000455441402279

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