Properties

Label 2-24e2-576.205-c1-0-6
Degree $2$
Conductor $576$
Sign $0.585 - 0.810i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 − 0.315i)2-s + (0.0432 − 1.73i)3-s + (1.80 + 0.870i)4-s + (0.456 + 0.520i)5-s + (−0.606 + 2.37i)6-s + (−4.15 − 0.547i)7-s + (−2.20 − 1.76i)8-s + (−2.99 − 0.149i)9-s + (−0.464 − 0.861i)10-s + (2.53 + 5.13i)11-s + (1.58 − 3.08i)12-s + (−0.126 − 0.371i)13-s + (5.56 + 2.06i)14-s + (0.920 − 0.767i)15-s + (2.48 + 3.13i)16-s + (0.495 + 0.495i)17-s + ⋯
L(s)  = 1  + (−0.974 − 0.223i)2-s + (0.0249 − 0.999i)3-s + (0.900 + 0.435i)4-s + (0.204 + 0.232i)5-s + (−0.247 + 0.968i)6-s + (−1.57 − 0.206i)7-s + (−0.780 − 0.625i)8-s + (−0.998 − 0.0498i)9-s + (−0.146 − 0.272i)10-s + (0.763 + 1.54i)11-s + (0.457 − 0.889i)12-s + (−0.0349 − 0.102i)13-s + (1.48 + 0.552i)14-s + (0.237 − 0.198i)15-s + (0.621 + 0.783i)16-s + (0.120 + 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475548 + 0.243087i\)
\(L(\frac12)\) \(\approx\) \(0.475548 + 0.243087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.315i)T \)
3 \( 1 + (-0.0432 + 1.73i)T \)
good5 \( 1 + (-0.456 - 0.520i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (4.15 + 0.547i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (-2.53 - 5.13i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (0.126 + 0.371i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (-0.495 - 0.495i)T + 17iT^{2} \)
19 \( 1 + (1.00 - 5.04i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-0.602 - 4.57i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-5.98 + 0.392i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (6.11 - 3.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.01 + 5.09i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.794 - 6.03i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (4.43 - 2.18i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (-4.02 + 1.07i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.71 - 8.55i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (-5.63 - 6.42i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (0.516 + 7.88i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (-7.22 - 3.56i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-0.125 - 0.303i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.86 - 9.32i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.653 + 2.43i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-2.99 - 2.62i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (8.36 - 3.46i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-9.41 - 5.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.62071519109913868717099239128, −9.836836989638408083207127606764, −9.289797783884208620385690417371, −8.164395587274902272022692288800, −7.15486277099149641048802460693, −6.71239527550275784115306512527, −5.90263779495962124047147143665, −3.76779311080841032923810828723, −2.65387517104976080055280243616, −1.45258937871516177972503928997, 0.41519430935816137151317758363, 2.76025430386618357009076285780, 3.59488595699283072816366825367, 5.26723294213467135296650753523, 6.20199450842703557527185848211, 6.81167310418421206991081802727, 8.427075372945116618300896715974, 9.027836757887522153219888811809, 9.492807783945436724209207958291, 10.40912993425230429054272462994

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