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Properties

Label 2-24e2-36.23-c1-0-12
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

\(L(1)\) \(\approx\) \(1.56703 + 0.137097i\)
\(L(\frac12)\) \(\approx\) \(1.56703 + 0.137097i\)
\(L(\frac{3}{2})\) 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

−10.63649720319680249865430626686, −10.34186580140632018668047736394, −9.184223038330498079473009340110, −8.074436506664780633222112159620, −6.96075759571043762402783946001, −5.91991095027876157514805336914, −5.46300293828809589109798412779, −4.25963847172732980825448104873, −2.87341808811382447272501028659, −1.22369546718801342396975393974, 1.50231020726581931292640182387, 2.13359376499004227954664019551, 4.61793509258135293692749458036, 5.18402986342618963934834912140, 5.96993288783695544809878473419, 6.93691878272699342381537590660, 8.100971969584353498748225622445, 8.957725836169782290548467459371, 9.867603382615333971772600237512, 10.74058731024643857336293671357

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