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Properties

Label 2-24e2-48.11-c3-0-15
Degree $2$
Conductor $576$
Sign $0.937 - 0.346i$
Analytic cond. $33.9851$
Root an. cond. $5.82967$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.62 + 3.62i)5-s + 33.7·7-s + (44.1 + 44.1i)11-s + (42.1 − 42.1i)13-s − 17.6i·17-s + (−99.2 − 99.2i)19-s + 83.8i·23-s + 98.7i·25-s + (−89.7 − 89.7i)29-s + 46.2i·31-s + (−122. + 122. i)35-s + (−7.47 − 7.47i)37-s + 299.·41-s + (56.3 − 56.3i)43-s + 280.·47-s + ⋯
L(s)  = 1  + (−0.324 + 0.324i)5-s + 1.82·7-s + (1.21 + 1.21i)11-s + (0.898 − 0.898i)13-s − 0.251i·17-s + (−1.19 − 1.19i)19-s + 0.759i·23-s + 0.789i·25-s + (−0.574 − 0.574i)29-s + 0.267i·31-s + (−0.590 + 0.590i)35-s + (−0.0332 − 0.0332i)37-s + 1.13·41-s + (0.199 − 0.199i)43-s + 0.869·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.346i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.937 - 0.346i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ 0.937 - 0.346i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.596471617\)
\(L(\frac12)\) \(\approx\) \(2.596471617\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 + (3.62 - 3.62i)T - 125iT^{2} \)
7 \( 1 - 33.7T + 343T^{2} \)
11 \( 1 + (-44.1 - 44.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (-42.1 + 42.1i)T - 2.19e3iT^{2} \)
17 \( 1 + 17.6iT - 4.91e3T^{2} \)
19 \( 1 + (99.2 + 99.2i)T + 6.85e3iT^{2} \)
23 \( 1 - 83.8iT - 1.21e4T^{2} \)
29 \( 1 + (89.7 + 89.7i)T + 2.43e4iT^{2} \)
31 \( 1 - 46.2iT - 2.97e4T^{2} \)
37 \( 1 + (7.47 + 7.47i)T + 5.06e4iT^{2} \)
41 \( 1 - 299.T + 6.89e4T^{2} \)
43 \( 1 + (-56.3 + 56.3i)T - 7.95e4iT^{2} \)
47 \( 1 - 280.T + 1.03e5T^{2} \)
53 \( 1 + (-8.15 + 8.15i)T - 1.48e5iT^{2} \)
59 \( 1 + (-193. - 193. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-127. + 127. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-110. - 110. i)T + 3.00e5iT^{2} \)
71 \( 1 - 1.07e3iT - 3.57e5T^{2} \)
73 \( 1 + 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 - 161. iT - 4.93e5T^{2} \)
83 \( 1 + (-64.4 + 64.4i)T - 5.71e5iT^{2} \)
89 \( 1 + 177.T + 7.04e5T^{2} \)
97 \( 1 - 559.T + 9.12e5T^{2} \)
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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

−10.65380925772209327486482503405, −9.357245858137399042602143034970, −8.582053022193636502741563121976, −7.64814218183137472055962887254, −6.98402687166812316106346511466, −5.68820232972221964694915144686, −4.63232854516021582700991368592, −3.87364150107803686234930704569, −2.21530091047345837839363132449, −1.13724059791649983276565645183, 1.00853623526455519124408793077, 1.95144242611764419877406690518, 3.91363655911059846680983382189, 4.36145545832585568997988014080, 5.73163234274964708234145050459, 6.52507773984156977906150523119, 7.907870344455352006259515683029, 8.525066471054619685014886950706, 9.002692100357846742351899769770, 10.59886391866156436732425840037

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