Properties

Label 2-24e2-576.205-c1-0-89
Degree $2$
Conductor $576$
Sign $0.149 - 0.988i$
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  + (0.390 − 1.35i)2-s + (−1.61 + 0.636i)3-s + (−1.69 − 1.06i)4-s + (−2.42 − 2.76i)5-s + (0.236 + 2.43i)6-s + (0.616 + 0.0811i)7-s + (−2.10 + 1.89i)8-s + (2.19 − 2.04i)9-s + (−4.70 + 2.21i)10-s + (−1.76 − 3.57i)11-s + (3.40 + 0.630i)12-s + (0.0634 + 0.186i)13-s + (0.350 − 0.806i)14-s + (5.66 + 2.91i)15-s + (1.75 + 3.59i)16-s + (−0.591 − 0.591i)17-s + ⋯
L(s)  = 1  + (0.275 − 0.961i)2-s + (−0.930 + 0.367i)3-s + (−0.847 − 0.530i)4-s + (−1.08 − 1.23i)5-s + (0.0964 + 0.995i)6-s + (0.232 + 0.0306i)7-s + (−0.743 + 0.668i)8-s + (0.730 − 0.683i)9-s + (−1.48 + 0.701i)10-s + (−0.531 − 1.07i)11-s + (0.983 + 0.181i)12-s + (0.0175 + 0.0518i)13-s + (0.0937 − 0.215i)14-s + (1.46 + 0.752i)15-s + (0.437 + 0.899i)16-s + (−0.143 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.149 - 0.988i$
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: \((2,\ 576,\ (\ :1/2),\ 0.149 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0377013 + 0.0324221i\)
\(L(\frac12)\) \(\approx\) \(0.0377013 + 0.0324221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.390 + 1.35i)T \)
3 \( 1 + (1.61 - 0.636i)T \)
good5 \( 1 + (2.42 + 2.76i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (-0.616 - 0.0811i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (1.76 + 3.57i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (-0.0634 - 0.186i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (0.591 + 0.591i)T + 17iT^{2} \)
19 \( 1 + (1.08 - 5.46i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-1.07 - 8.14i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (5.68 - 0.372i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (-1.29 + 0.746i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.255 + 1.28i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (0.984 + 7.47i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (-5.78 + 2.85i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (0.527 - 0.141i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.30 + 1.95i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (3.64 + 4.15i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.851 - 12.9i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (5.90 + 2.90i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-4.45 - 10.7i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.522 - 1.26i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.14 + 4.28i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (6.44 + 5.65i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (12.3 - 5.13i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (13.1 + 7.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.29449061453479869868339972423, −9.287385781222142606128723843039, −8.516889158537340998539672928036, −7.56178259645920300534531706153, −5.72118964439154171261300698395, −5.31469859334409763124196107753, −4.18262632214323882649615109857, −3.55769889004735840409096665964, −1.34897184657320523590038767224, −0.03160356923747754969286938108, 2.73648923631175865905899372350, 4.27984693987508014947352469741, 4.87537298345613036955827670912, 6.27404932569685206839528876834, 6.91668638586989601028971861735, 7.50988480341187065763146472749, 8.276059450895448932289448335648, 9.728760054331649162337339893063, 10.78044721274099122666988925020, 11.31105667194245186465732991135

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