Properties

Label 2-24e2-576.205-c1-0-54
Degree $2$
Conductor $576$
Sign $-0.731 + 0.681i$
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  + (−1.41 + 0.0559i)2-s + (−1.05 − 1.37i)3-s + (1.99 − 0.158i)4-s + (1.11 + 1.26i)5-s + (1.56 + 1.88i)6-s + (−0.644 − 0.0848i)7-s + (−2.80 + 0.335i)8-s + (−0.778 + 2.89i)9-s + (−1.64 − 1.73i)10-s + (−2.31 − 4.69i)11-s + (−2.31 − 2.57i)12-s + (1.12 + 3.31i)13-s + (0.915 + 0.0838i)14-s + (0.571 − 2.86i)15-s + (3.94 − 0.630i)16-s + (−2.81 − 2.81i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0395i)2-s + (−0.608 − 0.793i)3-s + (0.996 − 0.0790i)4-s + (0.498 + 0.567i)5-s + (0.639 + 0.768i)6-s + (−0.243 − 0.0320i)7-s + (−0.992 + 0.118i)8-s + (−0.259 + 0.965i)9-s + (−0.520 − 0.547i)10-s + (−0.698 − 1.41i)11-s + (−0.669 − 0.742i)12-s + (0.312 + 0.919i)13-s + (0.244 + 0.0224i)14-s + (0.147 − 0.740i)15-s + (0.987 − 0.157i)16-s + (−0.682 − 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.731 + 0.681i$
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: \((2,\ 576,\ (\ :1/2),\ -0.731 + 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162099 - 0.412062i\)
\(L(\frac12)\) \(\approx\) \(0.162099 - 0.412062i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0559i)T \)
3 \( 1 + (1.05 + 1.37i)T \)
good5 \( 1 + (-1.11 - 1.26i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (0.644 + 0.0848i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (2.31 + 4.69i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (-1.12 - 3.31i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (2.81 + 2.81i)T + 17iT^{2} \)
19 \( 1 + (-0.851 + 4.28i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-0.0561 - 0.426i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (0.430 - 0.0281i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (1.34 - 0.773i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.485 + 2.43i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (1.46 + 11.1i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (-3.21 + 1.58i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (11.7 - 3.14i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.72 - 4.08i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (5.86 + 6.69i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (0.622 + 9.49i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (1.70 + 0.841i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (0.383 + 0.925i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.871 - 2.10i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.586 + 2.18i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-6.77 - 5.94i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (2.69 - 1.11i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-6.88 - 3.97i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.65556505779555875491184222453, −9.435940072531012141299356279829, −8.641618953866130544295341435446, −7.66239575918884428219194704051, −6.70303304862957293538598698681, −6.27488009641268479758073138977, −5.19762028433230867058926059164, −3.05372206765359088644642435370, −2.03324835135664430061059335520, −0.36056891584646098719715057080, 1.59473371823928084961149222372, 3.18546978745545961727014157400, 4.65681596488972899572354717902, 5.66606655622778078371430525418, 6.47512078669373144892766385395, 7.69612757497854312624128081026, 8.593363879940844530989507369289, 9.621043249228316852007230229004, 10.01050946222653842199373169919, 10.72803118974489261902510880898

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