Properties

Label 2-24e2-8.5-c3-0-29
Degree $2$
Conductor $576$
Sign $-0.707 - 0.707i$
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  − 19.8i·5-s − 19.8·7-s + 48i·11-s − 79.5i·13-s + 42·17-s − 92i·19-s − 39.7·23-s − 271·25-s − 19.8i·29-s − 139.·31-s + 396i·35-s + 198. i·37-s + 6·41-s + 92i·43-s − 39.7·47-s + ⋯
L(s)  = 1  − 1.77i·5-s − 1.07·7-s + 1.31i·11-s − 1.69i·13-s + 0.599·17-s − 1.11i·19-s − 0.360·23-s − 2.16·25-s − 0.127i·29-s − 0.807·31-s + 1.91i·35-s + 0.884i·37-s + 0.0228·41-s + 0.326i·43-s − 0.123·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2992180457\)
\(L(\frac12)\) \(\approx\) \(0.2992180457\)
\(L(\frac{5}{2})\) 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 \( 1 + 19.8iT - 125T^{2} \)
7 \( 1 + 19.8T + 343T^{2} \)
11 \( 1 - 48iT - 1.33e3T^{2} \)
13 \( 1 + 79.5iT - 2.19e3T^{2} \)
17 \( 1 - 42T + 4.91e3T^{2} \)
19 \( 1 + 92iT - 6.85e3T^{2} \)
23 \( 1 + 39.7T + 1.21e4T^{2} \)
29 \( 1 + 19.8iT - 2.43e4T^{2} \)
31 \( 1 + 139.T + 2.97e4T^{2} \)
37 \( 1 - 198. iT - 5.06e4T^{2} \)
41 \( 1 - 6T + 6.89e4T^{2} \)
43 \( 1 - 92iT - 7.95e4T^{2} \)
47 \( 1 + 39.7T + 1.03e5T^{2} \)
53 \( 1 - 497. iT - 1.48e5T^{2} \)
59 \( 1 - 516iT - 2.05e5T^{2} \)
61 \( 1 - 358. iT - 2.26e5T^{2} \)
67 \( 1 - 524iT - 3.00e5T^{2} \)
71 \( 1 - 994.T + 3.57e5T^{2} \)
73 \( 1 + 430T + 3.89e5T^{2} \)
79 \( 1 + 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + 432iT - 5.71e5T^{2} \)
89 \( 1 + 630T + 7.04e5T^{2} \)
97 \( 1 - 862T + 9.12e5T^{2} \)
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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

−9.708870282534436276077902788277, −8.999178594735391316249340914410, −8.061942647200421762228461127057, −7.20128342874070722646190044949, −5.85855157752119952850807721658, −5.10549685591805685928974464264, −4.18019106788526052585929223640, −2.82889650211528723770220127447, −1.22067999261163554570076474870, −0.092508208861167735424008066167, 2.05584605896764347943284107815, 3.33119190085636159839446198485, 3.75734297713093941511841362275, 5.74571087730570693987680185279, 6.44322515407994678326487314230, 7.03679867617726707311363152128, 8.119702188887692381310879835778, 9.353958222036313183399609673598, 10.00355460302060978418938634548, 10.91526055091492777321411932290

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