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Properties

Label 2-24e2-48.11-c3-0-1
Degree $2$
Conductor $576$
Sign $-0.685 - 0.727i$
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  + (6.30 − 6.30i)5-s − 27.2·7-s + (4.03 + 4.03i)11-s + (37.9 − 37.9i)13-s + 79.8i·17-s + (−75.2 − 75.2i)19-s − 25.2i·23-s + 45.3i·25-s + (−107. − 107. i)29-s + 237. i·31-s + (−171. + 171. i)35-s + (210. + 210. i)37-s − 378.·41-s + (−191. + 191. i)43-s − 417.·47-s + ⋯
L(s)  = 1  + (0.564 − 0.564i)5-s − 1.46·7-s + (0.110 + 0.110i)11-s + (0.809 − 0.809i)13-s + 1.13i·17-s + (−0.908 − 0.908i)19-s − 0.228i·23-s + 0.362i·25-s + (−0.689 − 0.689i)29-s + 1.37i·31-s + (−0.828 + 0.828i)35-s + (0.933 + 0.933i)37-s − 1.44·41-s + (−0.678 + 0.678i)43-s − 1.29·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.685 - 0.727i$
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: \((2,\ 576,\ (\ :3/2),\ -0.685 - 0.727i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4008366214\)
\(L(\frac12)\) \(\approx\) \(0.4008366214\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.30 + 6.30i)T - 125iT^{2} \)
7 \( 1 + 27.2T + 343T^{2} \)
11 \( 1 + (-4.03 - 4.03i)T + 1.33e3iT^{2} \)
13 \( 1 + (-37.9 + 37.9i)T - 2.19e3iT^{2} \)
17 \( 1 - 79.8iT - 4.91e3T^{2} \)
19 \( 1 + (75.2 + 75.2i)T + 6.85e3iT^{2} \)
23 \( 1 + 25.2iT - 1.21e4T^{2} \)
29 \( 1 + (107. + 107. i)T + 2.43e4iT^{2} \)
31 \( 1 - 237. iT - 2.97e4T^{2} \)
37 \( 1 + (-210. - 210. i)T + 5.06e4iT^{2} \)
41 \( 1 + 378.T + 6.89e4T^{2} \)
43 \( 1 + (191. - 191. i)T - 7.95e4iT^{2} \)
47 \( 1 + 417.T + 1.03e5T^{2} \)
53 \( 1 + (-139. + 139. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-282. - 282. i)T + 2.05e5iT^{2} \)
61 \( 1 + (255. - 255. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-348. - 348. i)T + 3.00e5iT^{2} \)
71 \( 1 - 321. iT - 3.57e5T^{2} \)
73 \( 1 + 135. iT - 3.89e5T^{2} \)
79 \( 1 + 522. iT - 4.93e5T^{2} \)
83 \( 1 + (444. - 444. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.10e3T + 7.04e5T^{2} \)
97 \( 1 + 1.06e3T + 9.12e5T^{2} \)
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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

−10.40340850343564325305949896832, −9.837229104479519368288667057290, −8.893023225519597047066400652305, −8.236650062909987331819370025473, −6.74561450353657895312448410606, −6.20954661814796142672802195484, −5.21488933601562963296940242100, −3.89293943008702423711770750003, −2.89311952224366835432316683861, −1.37685984634782742398672965669, 0.11603238018666743104475836744, 1.96216291566113323767484297032, 3.15211508197452967222464238504, 4.06447128406130747043636719488, 5.64069365316698429149753961007, 6.43866344034877846161508692708, 6.96729682364861399905581338494, 8.315763886531798304069288166771, 9.418535577975738090450134640071, 9.805573859013370962548192250106

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