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Properties

Label 2-24e2-16.5-c3-0-27
Degree $2$
Conductor $576$
Sign $-0.751 - 0.659i$
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  + (−8.22 − 8.22i)5-s − 2.67i·7-s + (−45.2 − 45.2i)11-s + (35.3 − 35.3i)13-s + 72.4·17-s + (−19.4 + 19.4i)19-s + 139. i·23-s + 10.3i·25-s + (−66.0 + 66.0i)29-s − 188.·31-s + (−21.9 + 21.9i)35-s + (−84.0 − 84.0i)37-s + 104. i·41-s + (31.4 + 31.4i)43-s − 488.·47-s + ⋯
L(s)  = 1  + (−0.735 − 0.735i)5-s − 0.144i·7-s + (−1.23 − 1.23i)11-s + (0.755 − 0.755i)13-s + 1.03·17-s + (−0.234 + 0.234i)19-s + 1.26i·23-s + 0.0826i·25-s + (−0.422 + 0.422i)29-s − 1.09·31-s + (−0.106 + 0.106i)35-s + (−0.373 − 0.373i)37-s + 0.398i·41-s + (0.111 + 0.111i)43-s − 1.51·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.751 - 0.659i$
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: \((2,\ 576,\ (\ :3/2),\ -0.751 - 0.659i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1135813253\)
\(L(\frac12)\) \(\approx\) \(0.1135813253\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.22 + 8.22i)T + 125iT^{2} \)
7 \( 1 + 2.67iT - 343T^{2} \)
11 \( 1 + (45.2 + 45.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (-35.3 + 35.3i)T - 2.19e3iT^{2} \)
17 \( 1 - 72.4T + 4.91e3T^{2} \)
19 \( 1 + (19.4 - 19.4i)T - 6.85e3iT^{2} \)
23 \( 1 - 139. iT - 1.21e4T^{2} \)
29 \( 1 + (66.0 - 66.0i)T - 2.43e4iT^{2} \)
31 \( 1 + 188.T + 2.97e4T^{2} \)
37 \( 1 + (84.0 + 84.0i)T + 5.06e4iT^{2} \)
41 \( 1 - 104. iT - 6.89e4T^{2} \)
43 \( 1 + (-31.4 - 31.4i)T + 7.95e4iT^{2} \)
47 \( 1 + 488.T + 1.03e5T^{2} \)
53 \( 1 + (149. + 149. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-284. - 284. i)T + 2.05e5iT^{2} \)
61 \( 1 + (228. - 228. i)T - 2.26e5iT^{2} \)
67 \( 1 + (139. - 139. i)T - 3.00e5iT^{2} \)
71 \( 1 - 453. iT - 3.57e5T^{2} \)
73 \( 1 - 259. iT - 3.89e5T^{2} \)
79 \( 1 + 323.T + 4.93e5T^{2} \)
83 \( 1 + (563. - 563. i)T - 5.71e5iT^{2} \)
89 \( 1 - 866. iT - 7.04e5T^{2} \)
97 \( 1 + 936.T + 9.12e5T^{2} \)
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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

−9.829388680074660139615734261073, −8.608395570477204365219991943331, −8.121527480636378797103965743582, −7.36281430062404804135240485558, −5.76047319807483073499539817464, −5.31998204847991829736432232825, −3.87018718200702198935966504891, −3.09015463394317440452915097546, −1.19876379811931008539505127431, −0.03625185614507454039738692329, 1.94075529887901435059212776240, 3.13467187713724162990569767233, 4.21607134625113474293589198534, 5.26608170915060409375073276238, 6.50615255676973424175479014238, 7.36273359020413434667216270633, 8.013048344239276686750754795238, 9.111446591570111345848105371982, 10.16827805491973405074806275912, 10.79107660317201219400709932585

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