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Properties

Label 2-24e2-576.517-c1-0-39
Degree $2$
Conductor $576$
Sign $-0.586 - 0.809i$
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  + (−0.349 + 1.37i)2-s + (1.71 + 0.207i)3-s + (−1.75 − 0.958i)4-s + (−2.21 + 2.52i)5-s + (−0.885 + 2.28i)6-s + (4.47 − 0.588i)7-s + (1.92 − 2.06i)8-s + (2.91 + 0.712i)9-s + (−2.68 − 3.91i)10-s + (−1.40 + 2.85i)11-s + (−2.81 − 2.01i)12-s + (0.366 − 1.07i)13-s + (−0.757 + 6.33i)14-s + (−4.32 + 3.88i)15-s + (2.16 + 3.36i)16-s + (0.812 − 0.812i)17-s + ⋯
L(s)  = 1  + (−0.247 + 0.968i)2-s + (0.992 + 0.119i)3-s + (−0.877 − 0.479i)4-s + (−0.989 + 1.12i)5-s + (−0.361 + 0.932i)6-s + (1.68 − 0.222i)7-s + (0.681 − 0.731i)8-s + (0.971 + 0.237i)9-s + (−0.848 − 1.23i)10-s + (−0.424 + 0.860i)11-s + (−0.813 − 0.581i)12-s + (0.101 − 0.299i)13-s + (−0.202 + 1.69i)14-s + (−1.11 + 1.00i)15-s + (0.540 + 0.841i)16-s + (0.197 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.586 - 0.809i$
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: \((2,\ 576,\ (\ :1/2),\ -0.586 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736182 + 1.44274i\)
\(L(\frac12)\) \(\approx\) \(0.736182 + 1.44274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.349 - 1.37i)T \)
3 \( 1 + (-1.71 - 0.207i)T \)
good5 \( 1 + (2.21 - 2.52i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (-4.47 + 0.588i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (1.40 - 2.85i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (-0.366 + 1.07i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (-0.812 + 0.812i)T - 17iT^{2} \)
19 \( 1 + (-0.674 - 3.39i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (0.460 - 3.49i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (7.12 + 0.466i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (4.47 + 2.58i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.859 - 4.31i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-1.07 + 8.20i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-6.38 - 3.15i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (3.45 + 0.925i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.03 - 3.04i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-9.14 + 10.4i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.0872 - 1.33i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (12.5 - 6.19i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (2.10 - 5.08i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.01 - 4.86i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.56 + 13.3i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-12.2 + 10.7i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (-9.22 - 3.82i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (1.92 - 1.10i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.77561866591220473846835582213, −10.10700754301861662046335490899, −9.013348587935193914618187967657, −7.891913713078520830775050552246, −7.68673803369333996026890268147, −7.11812403409486501115677876609, −5.47422180731304885908130952763, −4.39210084984785294612494582892, −3.58933042049296331767383447981, −1.86249883184878747231558443335, 1.01332547965302537808983896757, 2.19655606447183263754297114916, 3.63035983281931376920985988680, 4.48214113990565730142539069581, 5.24112315251125647966804502191, 7.58508648918618314193657081690, 7.998699636109778621502538141249, 8.806603767565681827065950964427, 9.096419718684618802786187419972, 10.63871789804872699718529333176

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