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Properties

Label 2-24e2-16.5-c3-0-18
Degree $2$
Conductor $576$
Sign $0.840 + 0.541i$
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  + (4.66 + 4.66i)5-s − 24.8i·7-s + (22.3 + 22.3i)11-s + (−11.2 + 11.2i)13-s + 88.4·17-s + (−37.8 + 37.8i)19-s − 48.1i·23-s − 81.4i·25-s + (−10.4 + 10.4i)29-s + 96.9·31-s + (116. − 116. i)35-s + (−163. − 163. i)37-s − 360. i·41-s + (100. + 100. i)43-s + 220.·47-s + ⋯
L(s)  = 1  + (0.417 + 0.417i)5-s − 1.34i·7-s + (0.612 + 0.612i)11-s + (−0.240 + 0.240i)13-s + 1.26·17-s + (−0.456 + 0.456i)19-s − 0.436i·23-s − 0.651i·25-s + (−0.0668 + 0.0668i)29-s + 0.561·31-s + (0.560 − 0.560i)35-s + (−0.725 − 0.725i)37-s − 1.37i·41-s + (0.355 + 0.355i)43-s + 0.684·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.840 + 0.541i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.210083102\)
\(L(\frac12)\) \(\approx\) \(2.210083102\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.66 - 4.66i)T + 125iT^{2} \)
7 \( 1 + 24.8iT - 343T^{2} \)
11 \( 1 + (-22.3 - 22.3i)T + 1.33e3iT^{2} \)
13 \( 1 + (11.2 - 11.2i)T - 2.19e3iT^{2} \)
17 \( 1 - 88.4T + 4.91e3T^{2} \)
19 \( 1 + (37.8 - 37.8i)T - 6.85e3iT^{2} \)
23 \( 1 + 48.1iT - 1.21e4T^{2} \)
29 \( 1 + (10.4 - 10.4i)T - 2.43e4iT^{2} \)
31 \( 1 - 96.9T + 2.97e4T^{2} \)
37 \( 1 + (163. + 163. i)T + 5.06e4iT^{2} \)
41 \( 1 + 360. iT - 6.89e4T^{2} \)
43 \( 1 + (-100. - 100. i)T + 7.95e4iT^{2} \)
47 \( 1 - 220.T + 1.03e5T^{2} \)
53 \( 1 + (-175. - 175. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-405. - 405. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-664. + 664. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-107. + 107. i)T - 3.00e5iT^{2} \)
71 \( 1 + 215. iT - 3.57e5T^{2} \)
73 \( 1 - 668. iT - 3.89e5T^{2} \)
79 \( 1 - 822.T + 4.93e5T^{2} \)
83 \( 1 + (-326. + 326. i)T - 5.71e5iT^{2} \)
89 \( 1 + 262. iT - 7.04e5T^{2} \)
97 \( 1 + 150.T + 9.12e5T^{2} \)
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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

−10.23530788394792263514436112961, −9.642384910647262916426988445653, −8.419278736639223536459234581881, −7.35223947583001039569602742771, −6.80472974565639382340498433002, −5.72329385478231601704566167930, −4.41650918039507392720611277781, −3.62274624334156358078928724145, −2.13345846336532752118324263438, −0.796072023031706854653421869869, 1.10296005975909847114705600785, 2.43190673308079844446669745931, 3.56151082193274882812772300729, 5.09886607097291529477293488256, 5.67713437292410176546883780929, 6.61678519937092325769633908261, 7.932117089402941035928112254698, 8.759154265376704528633107022341, 9.392674543558759406254408179656, 10.25032495204282409529957059211

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