Properties

Label 2-24e2-4.3-c6-0-40
Degree $2$
Conductor $576$
Sign $i$
Analytic cond. $132.511$
Root an. cond. $11.5113$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 170.·5-s − 374. i·7-s − 1.35e3i·11-s + 2.95e3·13-s + 6.65e3·17-s − 9.92e3i·19-s + 1.55e4i·23-s + 1.35e4·25-s − 2.08e3·29-s + 2.84e4i·31-s + 6.39e4i·35-s + 3.33e3·37-s + 6.31e4·41-s + 8.85e4i·43-s − 1.08e4i·47-s + ⋯
L(s)  = 1  − 1.36·5-s − 1.09i·7-s − 1.02i·11-s + 1.34·13-s + 1.35·17-s − 1.44i·19-s + 1.28i·23-s + 0.866·25-s − 0.0854·29-s + 0.954i·31-s + 1.49i·35-s + 0.0657·37-s + 0.915·41-s + 1.11i·43-s − 0.104i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $i$
Analytic conductor: \(132.511\)
Root analytic conductor: \(11.5113\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3),\ i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.761942804\)
\(L(\frac12)\) \(\approx\) \(1.761942804\)
\(L(4)\) 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 \( 1 + 170.T + 1.56e4T^{2} \)
7 \( 1 + 374. iT - 1.17e5T^{2} \)
11 \( 1 + 1.35e3iT - 1.77e6T^{2} \)
13 \( 1 - 2.95e3T + 4.82e6T^{2} \)
17 \( 1 - 6.65e3T + 2.41e7T^{2} \)
19 \( 1 + 9.92e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.55e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.08e3T + 5.94e8T^{2} \)
31 \( 1 - 2.84e4iT - 8.87e8T^{2} \)
37 \( 1 - 3.33e3T + 2.56e9T^{2} \)
41 \( 1 - 6.31e4T + 4.75e9T^{2} \)
43 \( 1 - 8.85e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.08e4iT - 1.07e10T^{2} \)
53 \( 1 + 4.80e4T + 2.21e10T^{2} \)
59 \( 1 + 1.47e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.24e5T + 5.15e10T^{2} \)
67 \( 1 + 8.75e4iT - 9.04e10T^{2} \)
71 \( 1 + 2.25e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.75e5T + 1.51e11T^{2} \)
79 \( 1 - 2.76e5iT - 2.43e11T^{2} \)
83 \( 1 + 4.85e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.06e6T + 4.96e11T^{2} \)
97 \( 1 - 1.46e6T + 8.32e11T^{2} \)
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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

−9.494054475866198564375463276392, −8.458162354657759106811611314980, −7.78624148221532095355986810031, −7.06596152448687039743033467178, −5.95179762385792034713871626884, −4.73198495084620847833317270695, −3.61069831130173265115048383285, −3.31206173825885228773141004376, −1.14057746112059339642575522227, −0.53067283373581642788636392722, 0.886898202063767979063627561053, 2.21028368957670331789798615768, 3.51472697652565006101594003552, 4.19368126270512192906372566359, 5.45841341818688898699240135917, 6.31661764251046101090698813654, 7.58463553550660077735065885208, 8.134599550822065961146423656373, 8.937792083438958051377621456145, 10.01952194570701389841344417379

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