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Properties

Label 2-24e2-9.5-c2-0-32
Degree $2$
Conductor $576$
Sign $-0.452 + 0.891i$
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  + (−1.65 + 2.50i)3-s + (0.721 + 0.416i)5-s + (1.26 + 2.18i)7-s + (−3.53 − 8.27i)9-s + (−9.47 + 5.47i)11-s + (−4.36 + 7.55i)13-s + (−2.23 + 1.11i)15-s − 20.8i·17-s − 1.50·19-s + (−7.54 − 0.454i)21-s + (−1.00 − 0.578i)23-s + (−12.1 − 21.0i)25-s + (26.5 + 4.84i)27-s + (−15.7 + 9.08i)29-s + (−25.6 + 44.4i)31-s + ⋯
L(s)  = 1  + (−0.551 + 0.834i)3-s + (0.144 + 0.0832i)5-s + (0.180 + 0.311i)7-s + (−0.392 − 0.919i)9-s + (−0.861 + 0.497i)11-s + (−0.335 + 0.581i)13-s + (−0.148 + 0.0744i)15-s − 1.22i·17-s − 0.0790·19-s + (−0.359 − 0.0216i)21-s + (−0.0435 − 0.0251i)23-s + (−0.486 − 0.842i)25-s + (0.983 + 0.179i)27-s + (−0.542 + 0.313i)29-s + (−0.828 + 1.43i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.452 + 0.891i$
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: \((2,\ 576,\ (\ :1),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1420411536\)
\(L(\frac12)\) \(\approx\) \(0.1420411536\)
\(L(2)\) 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 \( 1 + (1.65 - 2.50i)T \)
good5 \( 1 + (-0.721 - 0.416i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.26 - 2.18i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (9.47 - 5.47i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.36 - 7.55i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 20.8iT - 289T^{2} \)
19 \( 1 + 1.50T + 361T^{2} \)
23 \( 1 + (1.00 + 0.578i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (15.7 - 9.08i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (25.6 - 44.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 7.93T + 1.36e3T^{2} \)
41 \( 1 + (21.8 + 12.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (19.3 + 33.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-59.6 + 34.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 46.5iT - 2.80e3T^{2} \)
59 \( 1 + (89.1 + 51.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (44.1 + 76.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-11.3 + 19.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 + 75.2T + 5.32e3T^{2} \)
79 \( 1 + (-51.8 - 89.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (53.7 - 31.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 1.95iT - 7.92e3T^{2} \)
97 \( 1 + (-59.2 - 102. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.26668110094478457831271369784, −9.474803956299502163672271152381, −8.719963498606355510801968166803, −7.45160377047705106175440696297, −6.53670025532777651554678777260, −5.30012726604039642451968128257, −4.84517637532556337948418045441, −3.53471684823887757337437459670, −2.20556674324524352417737642225, −0.05671497052860884516125591841, 1.45674825101403057450814225315, 2.76648720993658081120129798856, 4.29852454790744554785824223446, 5.63563136076709397318296211579, 6.00325392531820015225026556869, 7.50005355098402734638213927746, 7.78883779330419789963098271385, 8.920179527688782670562715014204, 10.19504846852235187081344063074, 10.85733369222221715752944393192

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