Properties

Label 2-24e2-9.5-c2-0-8
Degree $2$
Conductor $576$
Sign $-0.642 - 0.766i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + (3.39 + 1.96i)5-s + (6.39 + 11.0i)7-s + 9·9-s + (−14.2 + 8.25i)11-s + (−1.39 + 2.42i)13-s + (−10.1 − 5.88i)15-s − 2.54i·17-s + 21.5·19-s + (−19.1 − 33.2i)21-s + (−2.60 − 1.50i)23-s + (−4.79 − 8.31i)25-s − 27·27-s + (13.1 − 7.61i)29-s + (−13.6 + 23.5i)31-s + ⋯
L(s)  = 1  − 3-s + (0.679 + 0.392i)5-s + (0.914 + 1.58i)7-s + 9-s + (−1.29 + 0.750i)11-s + (−0.107 + 0.186i)13-s + (−0.679 − 0.392i)15-s − 0.149i·17-s + 1.13·19-s + (−0.914 − 1.58i)21-s + (−0.113 − 0.0652i)23-s + (−0.191 − 0.332i)25-s − 27-s + (0.455 − 0.262i)29-s + (−0.438 + 0.759i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.166717340\)
\(L(\frac12)\) \(\approx\) \(1.166717340\)
\(L(2)\) 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 \( 1 + 3T \)
good5 \( 1 + (-3.39 - 1.96i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-6.39 - 11.0i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (14.2 - 8.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1.39 - 2.42i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 2.54iT - 289T^{2} \)
19 \( 1 - 21.5T + 361T^{2} \)
23 \( 1 + (2.60 + 1.50i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-13.1 + 7.61i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (13.6 - 23.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 10.4T + 1.36e3T^{2} \)
41 \( 1 + (34.5 + 19.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (17.0 + 29.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (58.1 - 33.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 100. iT - 2.80e3T^{2} \)
59 \( 1 + (-5.29 - 3.05i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-20.3 - 35.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (54.4 - 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 52.8iT - 5.04e3T^{2} \)
73 \( 1 - 68.7T + 5.32e3T^{2} \)
79 \( 1 + (12.7 + 22.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-52.0 + 30.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 7.62iT - 7.92e3T^{2} \)
97 \( 1 + (-50.7 - 87.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.80474380459429535527004020595, −10.09509272837436488693636861390, −9.266006673082090343613607238400, −8.098248037509627934891383628075, −7.18088318192835932446386344212, −6.02252634840992423638958316782, −5.33997725320508341351148742371, −4.74419241719274156209079354953, −2.69983270747904738958380692984, −1.73585402323493009765656508283, 0.50961658483301262789072033359, 1.62341324005338842498036189102, 3.57985217097449987865529813238, 4.95056599524645833715040198035, 5.28927594871518790996437788931, 6.50579514993457588049681475747, 7.56373003156335229159321225932, 8.147228291421907288539073855138, 9.746607896835725010189276355939, 10.24689350225344678597778376385

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