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Properties

Label 2-24e2-36.23-c1-0-2
Degree $2$
Conductor $576$
Sign $-0.642 - 0.766i$
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  + 1.73i·3-s + (−1.5 − 0.866i)5-s + (1.5 − 0.866i)7-s − 2.99·9-s + (1.5 + 2.59i)11-s + (−2.5 + 4.33i)13-s + (1.49 − 2.59i)15-s + 6.92i·17-s − 3.46i·19-s + (1.49 + 2.59i)21-s + (−4.5 + 7.79i)23-s + (−1 − 1.73i)25-s − 5.19i·27-s + (−1.5 + 0.866i)29-s + (4.5 + 2.59i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.670 − 0.387i)5-s + (0.566 − 0.327i)7-s − 0.999·9-s + (0.452 + 0.783i)11-s + (−0.693 + 1.20i)13-s + (0.387 − 0.670i)15-s + 1.68i·17-s − 0.794i·19-s + (0.327 + 0.566i)21-s + (−0.938 + 1.62i)23-s + (−0.200 − 0.346i)25-s − 0.999i·27-s + (−0.278 + 0.160i)29-s + (0.808 + 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.642 - 0.766i$
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: \((2,\ 576,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418223 + 0.896882i\)
\(L(\frac12)\) \(\approx\) \(0.418223 + 0.896882i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
good5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 + 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.92iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.5 - 2.59i)T + (15.5 + 26.8i)T^{2} \)
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 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-7.5 + 4.33i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.05242509159075290161575453242, −10.10094412228770490708811384366, −9.364152863457945047263330864997, −8.484754992454991743751328371017, −7.65067692156559365177176818819, −6.52196270576756211609268904887, −5.19514060554776249485979679895, −4.30905149124184812142372682830, −3.79434331752425458044215031825, −1.91230205601250973957560956072, 0.55367377142151674232286080885, 2.36528936015397940407342549446, 3.36675424568723974781347826549, 4.92339147054828808244483384241, 5.93992809808174445966961616651, 6.88992913993431250234446206440, 7.952748910165349935676321261264, 8.159478430115781344256916246864, 9.447472471398254403865837540016, 10.61849591041226634927842687526

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