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Properties

Label 2-24e2-36.23-c1-0-11
Degree $2$
Conductor $576$
Sign $0.984 - 0.173i$
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  + (1.5 − 0.866i)3-s + (3 + 1.73i)5-s + (−3 + 1.73i)7-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s + (2 − 3.46i)13-s + 6·15-s + 1.73i·17-s + 1.73i·19-s + (−3 + 5.19i)21-s + (3.5 + 6.06i)25-s − 5.19i·27-s + (3 − 1.73i)29-s + (4.5 + 2.59i)33-s − 12·35-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (1.34 + 0.774i)5-s + (−1.13 + 0.654i)7-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s + (0.554 − 0.960i)13-s + 1.54·15-s + 0.420i·17-s + 0.397i·19-s + (−0.654 + 1.13i)21-s + (0.700 + 1.21i)25-s − 0.999i·27-s + (0.557 − 0.321i)29-s + (0.783 + 0.452i)33-s − 2.02·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23411 + 0.195459i\)
\(L(\frac12)\) \(\approx\) \(2.23411 + 0.195459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 + 0.866i)T \)
good5 \( 1 + (-3 - 1.73i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 1.73i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 - 2.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (-3 + 1.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.8iT - 89T^{2} \)
97 \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.14069559994496422658207913127, −10.04374173775015300804833092415, −9.076504975622358508676270055850, −8.219460303645119564646346064531, −6.87851483358594982965262773710, −6.42535037141320614319280356163, −5.53199019256473956643755450123, −3.62363196865782396133594269974, −2.76269522057821848468888476798, −1.77708389283447358557893073027, 1.42613130742504914712939237712, 2.87560496125806906661118029770, 3.93662465711349724506420647441, 5.02609093084749221952817661269, 6.20790185086092601676890831339, 6.96214460241739106284324130843, 8.463291896966235723150019959319, 9.078878130059815791317740130140, 9.686960115219519920153797334249, 10.31745065572463801169795924465

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