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Properties

Label 2-24e2-48.35-c3-0-20
Degree $2$
Conductor $576$
Sign $-0.672 + 0.739i$
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  + (−11.7 − 11.7i)5-s + 11.9·7-s + (36.9 − 36.9i)11-s + (20.4 + 20.4i)13-s − 81.1i·17-s + (−29.9 + 29.9i)19-s + 163. i·23-s + 151. i·25-s + (201. − 201. i)29-s − 43.1i·31-s + (−141. − 141. i)35-s + (100. − 100. i)37-s − 345.·41-s + (−326. − 326. i)43-s − 116.·47-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)5-s + 0.647·7-s + (1.01 − 1.01i)11-s + (0.436 + 0.436i)13-s − 1.15i·17-s + (−0.361 + 0.361i)19-s + 1.48i·23-s + 1.21i·25-s + (1.28 − 1.28i)29-s − 0.249i·31-s + (−0.681 − 0.681i)35-s + (0.444 − 0.444i)37-s − 1.31·41-s + (−1.15 − 1.15i)43-s − 0.361·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.739i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.672 + 0.739i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ -0.672 + 0.739i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.344785076\)
\(L(\frac12)\) \(\approx\) \(1.344785076\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.7 + 11.7i)T + 125iT^{2} \)
7 \( 1 - 11.9T + 343T^{2} \)
11 \( 1 + (-36.9 + 36.9i)T - 1.33e3iT^{2} \)
13 \( 1 + (-20.4 - 20.4i)T + 2.19e3iT^{2} \)
17 \( 1 + 81.1iT - 4.91e3T^{2} \)
19 \( 1 + (29.9 - 29.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 163. iT - 1.21e4T^{2} \)
29 \( 1 + (-201. + 201. i)T - 2.43e4iT^{2} \)
31 \( 1 + 43.1iT - 2.97e4T^{2} \)
37 \( 1 + (-100. + 100. i)T - 5.06e4iT^{2} \)
41 \( 1 + 345.T + 6.89e4T^{2} \)
43 \( 1 + (326. + 326. i)T + 7.95e4iT^{2} \)
47 \( 1 + 116.T + 1.03e5T^{2} \)
53 \( 1 + (-16.3 - 16.3i)T + 1.48e5iT^{2} \)
59 \( 1 + (-46.4 + 46.4i)T - 2.05e5iT^{2} \)
61 \( 1 + (-69.6 - 69.6i)T + 2.26e5iT^{2} \)
67 \( 1 + (157. - 157. i)T - 3.00e5iT^{2} \)
71 \( 1 + 690. iT - 3.57e5T^{2} \)
73 \( 1 + 799. iT - 3.89e5T^{2} \)
79 \( 1 - 763. iT - 4.93e5T^{2} \)
83 \( 1 + (940. + 940. i)T + 5.71e5iT^{2} \)
89 \( 1 - 660.T + 7.04e5T^{2} \)
97 \( 1 + 821.T + 9.12e5T^{2} \)
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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

−9.855249292599746548087365321553, −8.857980227138941706070022238051, −8.367343705862593909648590211588, −7.51522049619003960293147894262, −6.33654742021438949129844156668, −5.17232536535163397363771273848, −4.28585544884547346318835202273, −3.43021274085139824023598846219, −1.51955451451459118559301831620, −0.43574812475653051698829203837, 1.45368056316785174068878250056, 2.94301291054395281002540167595, 4.00454825181027349033414641991, 4.81283943830429222266675874877, 6.53359203912557267080317613209, 6.85246136282859933627089644957, 8.117525191606581624269718690126, 8.555917371438287857609117737768, 10.02931811619834818057155602116, 10.70356886634887754398776859808

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