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Properties

Label 2-24e2-576.517-c1-0-52
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: $\chi_{576} (517, \cdot )$
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

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

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